How did Paul die

Paul's death comes after the end of the Bible, so we can't be sure. 

One tradition says he was beheaded in Rome and the last we hear of him in the Bible is during his imprisonment in Rome. But he did have plans to eventually go to Spain and some say he was released and eventually made that missionary journey as well and died in Rome later during another imprisonment.

This would mean that his appeal to Caesar as a Roman citizen was ultimately successful. This is not unlikely especially when you consider his behavior in the Roman prison when it collapsed and during the shipwreck. Paul and his companions could have taken the opportunity to be free, but they said stayed saving the lives of their captors.

Paul's actions might have been seen not as a guilty man, but an honorable kne seeking the vindication only Caesar could offer to his reputation. 

And history is rarely as neat and tidy as we would like it to be.

 Everyone agrees that Paul was beheaded in Rome, and the  New Testament ends with Paul in Rome. It's very tempting to say that he simply died there, since we have no records beyond legend, but absence of evidence is not evidence of absence.