The best answer to this question is “we do not know who wrote the book of Hebrews.” It is an anonymous book and the earliest suggested author (Paul) is rarely defended today. This also means we know very little about when the book was written, the original audience, or where the author was when he wrote. This is remarkable since we can know all these things with a high level of certainty for many of Paul’s letters (especially Romans, 1-2 Corinthians or Galatians).
But there are a number of things we can know about the author of Hebrews from the book itself.
The author was a Jewish Christian, but undoubtedly a Hellenistic Jew. This would account for his detailed knowledge of the Hebrew Bible as well as his use of the Septuagint. The author regularly links verses together based on key words and themes to create chains of texts to support a his point. This method is found among the later rabbis and was probably the way a synagogue sermon was constructed.
The author was highly educated in both the Hebrew Bible and Hellenistic philosophy. Scholars often point out the similarities between the book of Hebrews and the writings of the Jewish philosopher, Philo of Alexandria (30 B.C. – A.D. 50). It is possible the author was educated in Alexandria since he reads the Hebrew Bible in ways which are similar to Philo. But as James Thompson observes, the parallels between Philo and Hebrews are insufficient evidence to prove the author knew Philo or his exegetical techniques (Hebrews, 24).
The author may have been in the second generation of the church. Hebrew 2:3 implies that the writer has received tradition from others who are the witnesses of Jesus. While this could imply an individual living in the 60’s who not from Judea, it may only mean the author was not a follower of Jesus until after the resurrection. On the other hand, the writer could be someone who heard the message of Jesus as a youth from of the now elderly apostles, allowing for a date well past A.D. 70.
The author was influenced by Paul. It is hard to imagine a writer in the first century who has not heard of Paul, so this does not require him to be a companion of Paul or even part of the Pauline circle (Barnabas or Timothy, for example). Unlike James, the author of Hebrews did not make a conscious effort to deal with similar topics as Paul (justification by faith) nor is there a direct reference to Paul (as in 2 Peter). Undoubtedly the writer read or heard Paul and knows his theology. This helps explain how a tradition of Pauline authorship might have developed in the first place.
None of these points should be controversial. As it turns out, there only a few people mentioned in the New Testament who fit this description. Other than Paul, the obvious candidate is Apollos since he is from Alexandria, was a well-educated Hellenistic Jew, and had some influence from Paul or Pauline theology. But there are other candidates: Barnabas, Aquila and Priscilla, or even Timothy have been suggested as as potential authors of the book.
How do these four points help us to make sense of the content of Hebrews? To what extent does the author’s anonymity hinder (or help?) our understanding of the book.