What evidence is there that Peter was a bishop in Rome? The Encyclopaedia Brittanica says 


  The claims that the church of Rome was founded by Peter or that he served as its first bishop are in dispute and rest on evidence that is not earlier than the middle or late 2nd century.


Where in the Bible does it say that Peter was a Bishop?  Also, are there  any non-biblical 1st-century historical accounts that mention his being the Bishop of Rome?
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. 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, although O’Malley finds the circumstantial evidence is persuasive. Part of the evidence O'Malley relies on is in 1 Peter, which he acknowledges might not have been written by Peter himself, but claims it was at least written under his inspiration.

Francis A. Sullivan S.J. says, in From Apostles to Bishops, page 15, that there is a broad consensus among scholars, including most Catholic ones,  that the church of Rome was most probably led by a college of presbyters until well into the second century. He says there is no evidence of a ruling bishop in the first century. Now, it seems inconceivable that Peter would lead the church in Rome, as its first bishop, and appoint a successor as bishop, only for the Christian community of Rome to ignore this precedent for up to a century after the death of Linus.