1) Rebuttal to writing style - Scholars of Greek consider its writing to be more polished and eloquent than any other book of the New Testament, and "the very carefully composed and studied Greek of Hebrews is not Paul's spontaneous, volatile contextual Greek". Wikipedia Encyclopedia.
a) Paul was highly educated in Judaism.
If anyone else thinks he may have confidence in the flesh, I more so: circumcised the eighth day, of the stock of Israel, of the tribe of Benjamin, a Hebrew of the Hebrews; concerning the law, a Pharisee; concerning zeal, persecuting the church; concerning the righteousness which is in the law, blameless. Philippians 3:4-6
“Brethren and fathers, hear my defense before you now.” And when they heard that he spoke to them in the Hebrew language, they kept all the more silent. Then he said: “I am indeed a Jew, born in Tarsus of Cilicia, but brought up in this city at the feet of Gamaliel, taught according to the strictness of our fathers’ law, and was zealous toward God as you all are today.” Acts 22:1-3
b) Paul had different writers for his epistles.
I, Tertius, who wrote this epistle, greet you in the Lord. Romans 16:22
c) Paul was a favored prisoner while in Caesarea.
So, he commanded the centurion to keep Paul and to let him have liberty, and told him not to forbid any of his friends to provide for or visit him. And after some days, when Felix came with his wife Drusilla, who was Jewish, he sent for Paul and heard him concerning the faith in Christ. Acts 24:23-24
So, the next day, when Agrippa and Bernice had come with great pomp, and had entered the auditorium with the commanders and the prominent men of the city, at Festus’ command Paul was brought in. Acts 25:23
d) Paul had several years of confinement in Caesarea. This gave him time to compose his epistle to the Hebrews with help from court recorders.
But after two years Porcius Festus succeeded Felix; and Felix, wanting to do the Jews a favor, left Paul bound. Acts 24:27
 
2) Purpose of Paul writing to Hebrews.
a) Paul discovered that Jewish Christians in Jerusalem were still zealous for the Old Covenant.
And when we had come to Jerusalem, the brethren received us gladly. On the following day Paul went in with us to James, and all the elders were present. When he had greeted them, he told in detail those things which God had done among the Gentiles through his ministry. And when they heard it, they glorified the Lord. And they said to him, “You see, brother, how many myriads of Jews there are who have believed, and they are all zealous for the law; but they have been informed about you that you teach all the Jews who are among the Gentiles to forsake Moses, saying that they ought not to circumcise their children nor to walk according to the customs. Acts 21:17-21
b) Paul wanted to fully deliver his message of Jesus Christ to the non-believing Jews in Jerusalem.
And they listened to him until this word, and then they raised their voices and said, “Away with such a fellow from the earth, for he is not fit to live!” Acts 22:22
Then Paul, looking earnestly at the council, said, “Men and brethren, I have lived in all good conscience before God until this day.” And the high priest Ananias commanded those who stood by him to strike him on the mouth. Acts 23:1-2
c) Paul wanted to remind the Hebrews that faith instead of the law was the means of salvation.
Now faith is the substance of things hoped for, the evidence of things not seen. Hebrews 11:1
But before faith came, we were kept under guard by the law, kept for the faith which would afterward be revealed. Therefore, the law was our tutor to bring us to Christ, that we might be justified by faith. But after faith has come, we are no longer under a tutor. Galatians 3:23-25
d) Paul yearned for his fellow countrymen to be saved.
I tell the truth in Christ, I do not lie, my conscience also bearing me witness in the Holy Spirit, that I have great heaviness and continual pain in my heart. For I myself was wishing to be accursed from Christ for my brothers, my kinsmen according to the flesh, who are Israelites; to whom belong the adoption, and the glory, and the covenants, and the giving of the Law, and the service of God, and the promises; whose are the fathers, and of whom is the Christ according to flesh, He being God over all, blessed forever. Amen. Romans 9:1-5
 
3) Why did Paul not identify himself as the author?
a) Paul wanted believing Jews to read the epistle without bias.
Therefore, we must give the more earnest heed to the things we have heard, lest we drift away. Hebrews 2:1
b) Paul wanted non-believing Jews to read the epistle without bias.
And when it was day, some of the Jews banded together and bound themselves under an oath, saying that they would neither eat nor drink till they had killed Paul. Acts 23:12
c) Paul deferred to the original apostles so that his identity would be masked.
For if the word spoken through angels proved steadfast, and every transgression and disobedience received a just reward, how shall we escape if we neglect so great a salvation, which at the first began to be spoken by the Lord, and was confirmed to us by those who heard Him, God also bearing witness both with signs and wonders, with various miracles, and gifts of the Holy Spirit, according to His own will? Hebrews 2:2-4
 
4) Similar words used by Paul in his epistles.
a) has in these last days spoken to us by His Son, whom He has appointed heir of all things, through whom also He made the worlds; Hebrews 1:2
concerning His Son Jesus Christ our Lord, who was born of the seed of David according to the flesh, Romans 1:3
God is faithful, by whom you were called into the fellowship of His Son, Jesus Christ our Lord. 1 Corinthians 1:9
to reveal His Son in me, that I might preach Him among the Gentiles, I did not immediately confer with flesh and blood, Galatians 1:16
and to wait for His Son from heaven, whom He raised from the dead, even Jesus who delivers us from the wrath to come. 1 Thessalonians 1:10
b) who being the brightness of His glory and the express image of His person, Hebrews 1:3 and that He might make known the riches of His glory on the vessels of mercy, which He had prepared beforehand for glory, Romans 9:23 that we who first trusted in Christ should be to the praise of His glory. Ephesians 1:12
c) and upholding all things by the word of His power, Hebrews 1:3
What if God, wanting to show His wrath and to make His power known, endured with much longsuffering the vessels of wrath prepared for destruction, Romans 9:22
And God both raised up the Lord and will also raise us up by His power. 1 Corinthians 6:14
and what is the exceeding greatness of His power toward us who believe, according to the working of His mighty of His mighty power Ephesians 1:19
These shall be punished with everlasting destruction from the presence of the Lord and from the glory of His power, 2 Thessalonians 1:9
d) Seeing then that we have a great High Priest who has passed through the heavens, Jesus the Son of God, Hebrews 4:14
and declared to be the Son of God with power according to the Spirit of holiness, by the resurrection from the dead. Romans 1:4
For the Son of God, Jesus Christ, who was preached among you by us—by me, Silvanus, and Timothy—was not      Yes and No, but in Him was Yes.     2 Corinthians 1:19
I have been crucified with Christ; it is no longer I who live, but Christ lives in me; and the life which I now live in the flesh I live by faith in the Son of God, who loved me and gave Himself for me. Galatians 2:20
till we all come to the unity of the faith and of the knowledge of the Son of God, to a perfect man, to the measure of the stature of the fullness of Christ; Ephesians 4:13
e) Let us therefore come boldly to the throne of grace, that we may obtain mercy and find grace to help in time of need. Hebrews 4:16
Romans (20); 1 Corinthians (5); 2 Corinthians (12); Galatians (7); Ephesians (13); Philippians (3); Colossians (7); 1 Thessalonians (2); 2 Thessalonians (4); 1 Timothy (4); 2 Timothy (4); Titus (5); Philemon (2) Grace is used an additional 6 times in Hebrews.
f) And for this reason, He is the Mediator of the new covenant, by means of death, for the redemption of the transgressions under the first covenant, that those who are called may receive the promise of the eternal inheritance. Hebrews 9:15 Promise is used an additional 16 times in Hebrews.
Romans (10); 2 Corinthians (3); Galatians (11); Ephesians (4); 1 Timothy (1); 2 Timothy (1); Titus (1);
g) Therefore, since a promise remains of entering His rest, let us fear lest any of you seem to have come short of it. For indeed the gospel was preached to us as well as to them; but the word which they heard did not profit them, not being mixed with faith in those who heard it. Hebrews 4:1-2
Romans (38); 1 Corinthians (7); 2 Corinthians (7); Galatians (20); Ephesians (8); Philippians (5); Colossians (5); 1 Thessalonians (8); 2 Thessalonians (4); 1 Timothy (18); 2 Timothy (8); Titus (5); Philemon (2) Faith is used an additional 30 times in Hebrews.
 
5) Unbelieving Jews bitterly hated Paul for years due to the Hebrews letter.
Now after five days Ananias the high priest came down with the elders and a certain orator named Tertullus. These gave evidence to the governor against Paul. And when he was called upon, Tertullus began his accusation, saying: “Seeing that through you we enjoy great peace, and prosperity is being brought to this nation by your foresight, we accept it always and in all places, most noble Felix, with all thankfulness. Nevertheless, not to be tedious to you any further, I beg you to hear, by your courtesy, a few words from us. For we have found this man a plague, a creator of dissension among all the Jews throughout the world, and a ringleader of the sect of the Nazarenes. He even tried to profane the temple, and we seized him, and wanted to judge him according to our law. But the commander Lysias came by and with great violence took him out of our hands, commanding his accusers to come to you. By examining him yourself you may ascertain all these things of which we accuse him.” And the Jews also assented, maintaining that these things were so. Acts 24:1-9
But after two years Porcius Festus succeeded Felix; and Felix, wanting to do the Jews a favor, left Paul bound. Acts 24:27
Now when Festus had come to the province, after three days he went up from Caesarea to Jerusalem. Then the high priest and the chief men of the Jews informed him against Paul; and they petitioned him, asking a favor against him, that he would summon him to Jerusalem—while they lay in ambush along the road to kill him. But Festus answered that Paul should be kept at Caesarea, and that he himself was going there shortly. “Therefore,” he said, “let those who have authority among you go down with me and accuse this man, to see if there is any fault in him.” And when he had remained among them more than ten days, he went down to Caesarea. And the next day, sitting on the judgment seat, he commanded Paul to be brought. When he had come, the Jews who had come down from Jerusalem stood about and laid many serious complaints against Paul, which they could not prove, while he answered for himself, “Neither against the law of the Jews, nor against the temple, nor against Caesar have I offended in anything at all.” But Festus, wanting to do the Jews a favor, answered Paul and said, “Are you willing to go up to Jerusalem and there be judged before me concerning these things?” So Paul said, “I stand at Caesar’s judgment seat, where I ought to be judged. To the Jews I have done no wrong, as you very well know. For if I am an offender, or have committed anything deserving of death, I do not object to dying; but if there is nothing in these things of which these men accuse me, no one can deliver me to them. I appeal to Caesar.” Then Festus, when he had conferred with the council, answered, “You have appealed to Caesar? To Caesar you shall go!” Acts 25:1-12
 
6) Paul was in prison and his experiences were recorded.
But recall the former days in which, after you were illuminated, you endured a great struggle with sufferings: partly while you were made a spectacle both by reproaches and tribulations, and partly while you became companions of those who were so treated; for you had compassion on me in my chains, and joyfully accepted the plundering of your goods, knowing that you have a better and an enduring possession for yourselves in heaven. Hebrews 10:32-34
But at midnight Paul and Silas were praying and singing hymns to God, and the prisoners were listening to them. Suddenly there was a great earthquake, so that the foundations of the prison were shaken; and immediately all the doors were opened and everyone’s chains were loosed. Acts 16:25-26
And see, now I go bound in the spirit to Jerusalem, not knowing the things that will happen to me there, except that the Holy Spirit testifies in every city, saying that chains and tribulations await me. But none of these things move me; nor do I count my life dear to myself, so that I may finish my race with joy, and the ministry which I received from the Lord Jesus, to testify to the gospel of the grace of God. Acts 20:22-24
He immediately took soldiers and centurions, and ran down to them. And when they saw the commander and the soldiers, they stopped beating Paul. Then the commander came near and took him, and commanded him to be bound with two chains; and he asked who he was and what he had done. Acts 21:32-33
Then Agrippa said to Paul, “You almost persuade me to become a Christian.” And Paul said, “I would to God that not only you, but also all who hear me today, might become both almost and altogether such as I am, except for these chains.” When he had said these things, the king stood up, as well as the governor and Bernice and those who sat with them; and when they had gone aside, they talked among themselves, saying, “This man is doing nothing deserving of death or chains.” Acts 26:28-31
for which I am an ambassador in chains; that in it I may speak boldly, as I ought to speak. Ephesians 6:20
But I want you to know, brethren, that the things which happened to me have actually turned out for the furtherance of the gospel, so that it has become evident to the whole palace guard, and to all the rest, that my chains are in Christ; and most of the brethren in the Lord, having become confident by my chains, are much more bold to speak the word without fear. Some indeed preach Christ even from envy and strife, and some also from goodwill: The former preach Christ from selfish ambition, not sincerely, supposing to add affliction to my chains; but the latter out of love, knowing that I am appointed for the defense of the gospel. Philippians 1:12-16
Continue earnestly in prayer, being vigilant in it with thanksgiving; meanwhile praying also for us, that God would open to us a door for the word, to speak the mystery of Christ, for which I am also in chains, that I may make it manifest, as I ought to speak. Colossians 4:2-4
This salutation by my own hand—Paul. Remember my chains. Grace be with you. Amen. Colossians 4:18
Remember that Jesus Christ, of the seed of David, was raised from the dead according to my gospel, for which I suffer trouble as an evildoer, even to the point of chains; but the word of God is not chained. 2 Timothy 2:9
Therefore, though I might be very bold in Christ to command you what is fitting, yet for love’s sake I rather appeal to you—being such a one as Paul, the aged, and now also a prisoner of Jesus Christ— I appeal to you for my son Onesimus, whom I have begotten while in my chains, who once was unprofitable to you, but now is profitable to you and to me. I am sending him back. You therefore receive him, that is, my own heart, whom I wished to keep with me, that on your behalf he might minister to me in my chains for the gospel. Philemon 1:8-13
 
7) Paul and Timothy worked together in preaching the Gospel.
Know that our brother Timothy has been set at liberty, with whom, if he comes shortly, I will see you. Hebrews 13:23
And I appeal to you, brethren, bear with the word of exhortation, for I have written to you in few words. Know that our brother Timothy has been set free, with whom I shall see you if he comes shortly. Greet all those who rule over you, and all the saints. Those from Italy greet you. Grace be with you all. Amen. Acts 13:22-25
Then he came to Derbe and Lystra. And behold, a certain disciple was there, named Timothy, the son of a certain Jewish woman who believed, but his father was Greek. He was well spoken of by the brethren who were at Lystra and Iconium. Paul wanted to have him go on with him. And he took him and circumcised him because of the Jews who were in that region, for they all knew that his father was Greek. Acts 16:1-3
When Silas and Timothy had come from Macedonia, Paul was compelled by the Spirit, and testified to the Jews that Jesus is the Christ. Acts 18:5
Timothy, my fellow worker, and Lucius, Jason, and Sosipater, my countrymen, greet you. Romans 16:21
Therefore, I urge you, imitate me. For this reason I have sent Timothy to you, who is my beloved and faithful son in the Lord, who will remind you of my ways in Christ, as I teach everywhere in every church. 1 Corinthians 4:16-17
But as God is faithful, our word to you was not Yes and No. For the Son of God, Jesus Christ, who was preached among you by us—by me, Silvanus, and Timothy—was not Yes and No, but in Him was Yes. For all the promises of God in Him are Yes, and in Him Amen, to the glory of God through us. 2 Corinthians 1:18-20
Paul and Timothy, bondservants of Jesus Christ, to all the saints in Christ Jesus who are in Philippi, with the bishops and deacons: Grace to you and peace from God our Father and the Lord Jesus Christ. Philippians 1:1-2
Paul, an apostle of Jesus Christ by the will of God, and Timothy our brother, to the saints and faithful brethren in Christ who are in Colosse: Grace to you and peace from God our Father and the Lord Jesus Christ. Colossians 1:1-2
But now that Timothy has come to us from you, and brought us good news of your faith and love, and that you always have good remembrance of us, greatly desiring to see us, as we also to see you— therefore, brethren, in all our affliction and distress we were comforted concerning you by your faith. 1 Thessalonians 3:6-7
Paul, Silvanus, and Timothy, to the church of the Thessalonians in God our Father and the Lord Jesus Christ: Grace to you and peace from God our Father and the Lord Jesus Christ. 2 Thessalonians 1-2
O Timothy! Guard what was committed to your trust, avoiding the profane and idle babblings and contradictions of what is falsely called knowledge— by professing it some have strayed concerning the faith. Grace be with you. Amen. 1 Timothy 6:20-21
Paul, an apostle of Jesus Christ by the will of God, according to the promise of life which is in Christ Jesus, to Timothy, a beloved son: Grace, mercy, and peace from God the Father and Christ Jesus our Lord. I thank God, whom I serve with a pure conscience, as my forefathers did, as without ceasing I remember you in my prayers night and day, greatly desiring to see you, being mindful of your tears, that I may be filled with joy, when I call to remembrance the genuine faith that is in you, which dwelt first in your grandmother Lois and your mother Eunice, and I am persuaded is in you also. 2 Timothy 1:1-5