Peter’s death occurred around 66 AD. It is not recorded in the book of Acts, whose narrative only continues into 50s of the first century. It is also not explicitly mentioned in any of the New Testament letters, which were probably all written before 66 AD. There are, however, two allusions to Peter’s death.
One is written by Peter himself in 2 Peter 1:12-15:
Therefore I intend always to remind you of these qualities, though you know them and are established in the truth that you have. I think it right, as long as I am in this body, to stir you up by way of reminder, since I know that the putting off of my body will be soon, as our Lord Jesus Christ made clear to me. And I will make every effort so that after my departure you may be able at any time to recall these things.
This letter was probably written in the 60s of the first century at a time when Peter was anticipating his death. It may be that he was simply growing old and assuming that he would soon die naturally. Or he may have perceived the increasing threats on Christians and realised that soon his life would be threatened. The historical information we have outside the Bible indicates that Peter was crucified.
The other reference to Peter’s death is at the end of John’s gospel:
[Jesus said] “Truly, truly, I say to you, when you were young, you used to dress yourself and walk wherever you wanted, but when you are old, you will stretch out your hands, and another will dress you and carry you where you do not want to go.” (This he said to show by what kind of death he was to glorify God.) And after saying this he said to him, “Follow me.” (John 21:18-19)
The reference to Peter stretching out his hands would be consistent with the idea that Peter was crucified.