The Bible does not mention Peter as ever going to Rome, and there is no early Christian record of this being the case. Even at the end of the first century, the author of 1 Clement appears unaware that St. Peter ever came to Rome. Although written from Rome, 1 Clement mentions Peter's 'many labours' and makes a general comment about Peter's death, without mentioning Rome:
There was Peter who by reason of unrighteous jealousy endured not one not one but many labours, and thus having borne his testimony went to his appointed place of glory."
Rex Wyler, in The Jesus Sayings, page 252, says the legend that Peter visited Rome appears in the non-canonical Acts of Peter, composed in about 185 CE. Eusebius embellishes this report two centuries later by adding that both Peter and Paul were executed in Rome during Nero's persecutions, but he cites no source and gives an erroneous date, casting doubt on his research.
John W. O’Malley, S.J. says, in A History of the Popes, page 8, that no one piece of evidence states in straightforward and unambiguous language either that Peter either went to Rome or that he died there. However, he relies on circumstantial evidence, particularly First Peter, which he sees as suggesting, or even indicating, by its reference to 'Babylon' that Peter is in Rome at the time the letter was written, which was probably about the year 63:
1 Peter 5:12-13: "I write you this briefly through Silvanus, 6 whom I consider a faithful brother, exhorting you and testifying that this is the true grace of God. Remain firm in it. The chosen one at Babylon sends you greeting, as does Mark, my son."
O’Malley acknowledges that 1 Peter was likely not written by Peter himself, although he believes it was written under his inspiration, but surprisingly overlooks that the reference to ‘Babylon’ is an anachronism before 70 CE*. This fact points to First Peter as pseudepigraphical, and without this letter, Father O'Malley has little more than Catholic tradition that Peter went to Rome or was martyred there.
(*) Bart D. Ehrman says in Forged, page 68, scholars have long realised what the reference to ‘Babylon’ means in 1 Peter 5:13. Babylon was the city that had defeated Judah and destroyed Jerusalem and its Temple in the sixth century BCE. By the end of the first century Christians and Jews had started using the word Babylon as a code word for Rome, the city that was the enemy of God in their own day, which also destroyed Jerusalem and its Temple in the year 70. The author is claiming to be writing from the city of Rome, but using the code 'Babylon' is an anachronism during the lifetime of St. Peter.