It is generally assumed that Paul was killed under Nero around 67 AD although some think he could had died around 64 or 65 AD. Eusebius, in his Ecclesiastical History (325 AD) wrote in Book 2.25.6,8:
It is, therefore, recorded that Paul was beheaded in Rome itself, and that Peter likewise was crucified under Nero. This account of Peter and Paul is substantiated by the fact that their names are preserved in the cemeteries of that place even to the present day....
And that they both suffered martyrdom at the same time is stated by Dionysius, bishop of Corinth, in his epistle to the Romans, in the following words: “You have thus by such an admonition bound together the planting of Peter and of Paul at Rome and Corinth. For both of them planted and likewise taught us in our Corinth. And they taught together in like manner in Italy, and suffered martyrdom at the same time.” I have quoted these things in order that the truth of the history might be still more confirmed.
Eusebius quotes Dionysius' letter to the Romans (~180 AD), which we have only have segments of it, but what we do have mentions:
Therefore you also have by such admonition joined in close union the churches that were planted by Peter and Paul, that of the Romans and that of the Corinthians: for both of them went to our Corinth, and taught us in the same way as they taught you when they went to Italy; and having taught you, they suffered martyrdom at the same time.Which sounds a bit like the Eusbius quote. Also First Clement (88-97 AD) 5:1-9 mentions Peter and Paul's death, but not in any great detail.
The thing is, Acts doesn't mention the death of Paul at all. Like the destruction of Jerusalem, this is a huge deal that is left out, as the last half of Acts is all about Paul. Acts mentions Paul a whopping 132 times which by contrast, Jesus is only mentioned 72 times. Needless to say the death of Paul probably should have gotten a look in, but instead Acts ends almost abruptly with Paul under house arrest in Rome, where he was able to continue with his teachings. If Luke was written after 70 AD and Acts was written after it, there is no good reason as to why Luke would stop his account with Paul in prison.
It's also not like Luke doesn't avoid the death of Christians. Luke mentions in passing James the brother of John being killed by Herod and he goes into detail around the of the stoning of Stephen, who is not a big player and isn't mentioned in any other New Testament writings. If Luke and Acts were written after 70 AD it seems we are meant to believe that Luke skipped over the death of the main character of his book who was also the personal travelling companion of him.
The abrupt ending to Acts is remarkable and uncharacteristic of Luke’s thorough style. If Luke wrote after the outcome of Paul’s trial and failed to describe that outcome, it would be in some respects similar to his gospel account ending with Jesus being delivered to Pilate without any indication of his trial, his flogging, and his suffering, or his disciples claiming that he rose again. Luke gives no indication whatsoever of the slaughter and massacre of Christians under the Neronian persecution of AD 64. (p96)