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.
Where in the Bible does it say that Peter was a Bishop?
The English word "bishop" is commonly translated from the Greek word ἐπίσκοπος (episkopos). The noun ἐπίσκοπος literally means "one who watches over," i.e. overseer. In the New Testament, the ἐπίσκοπος was essentially synonymous with the "elder," or Greek πρεσβύτερος (presbyteros), as demonstrated by Acts 20:
17 And from Miletus he sent to Ephesus, and called the elders (πρεσβυτέρους) of the church. 28 Take heed therefore unto yourselves, and to all the flock, over the which the Holy Ghost hath made you overseers (ἐπισκόπους), to feed the church of God, which he hath purchased with his own blood. (KJV)
In Peter's first epistle, he declares that he was a πρεσβύτερος, and therefore, an ἐπίσκοπος.
5:1 The elders (Πρεσβυτέρους) which are among you I exhort, who am also an elder (συμπρεσβύτερος), and a witness of the sufferings of Christ, and also a partaker of the glory that shall be revealed: 5:2 Feed the flock of God which is among you, taking the oversight (ἐπισκοποῦντες) thereof, not by constraint, but willingly; not for filthy lucre, but of a ready mind; (KJV)
Notice how the apostle Peter, writing to the elders of the Church, refers to himself as a "fellow elder" (συμπρεσβύτερος). He then commands the elders to "feed the flock of God" (something which the apostle Paul described as a function of the elders/bishops in Acts 20:17/28) and take oversight, which is the word ἐπισκοπέω, the verb form of the noun ἐπίσκοπος.
In conclusion, yes, the apostle Peter was an elder and a bishop in the Church.
Also, are there any non-biblical 1st-century historical accounts that mention his being the Bishop of Rome?
As far as I know, there are no 1st century A.D. historical accounts that mention the apostle Peter being Bishop of Rome.
"Was St. Peter a bishop of Rome?"
This question will remain a bone of contention between those who believe he was and those who do not believe, as well as to what evidence one is willing to accept on this subject.
The only Biblical information we have that St. Peter might have been in Rome is the cryptic reference of 1 Peter 5:13, "She who is in Babylon, chosen together with you, sends you her greetings, and so does my son Mark." If "Babylon" is the cryptic name for Rome, as is commonly claimed, then this might be the sole source of information from the Bible.
I do not want to repeat what the others have to say about whether St. Peter was ever in Rome or not, so I will approach this question from a different angle.
Historical data for evidence that St. Peter was at Rome will remain slim, but that does not mean that his sojourn at Rome was none the less quite possible. Catholic tradition is too strong to simply put aside.
There exists in Rome a church dedicated to Our Lady under the title of St. Mary in Palmis, but its' original title was the Church of Domine Quo Vadis.
The title Domine Quo Vadis is based on the pious story that St. Peter in order to avoid the persecutions in Rome, decided to flee the city, but was met on way by Our Lord. He asked: Domine, quo vadis? ("Lord, where are you going?"), and Christ answered: "To Rome, to be crucified again." This helped Peter overcome his fear of martyrdom, and he returned to face his persecutors.
Quo Vadis or Domine, quo vadis?, meaning Lord, where are you going?, a text from the Apocryphal Acts of Peter composed c. a. d. 190, probably in Syria or Palestine. An anecdote based on the text became a legend in patristic times and is referred to by origen (Comm. in Joan. 20.12; Patrologia Graeca 14:600) and ambrose of milan (Sermo Contra Auxentium 13).
Peter is represented in flight from Rome during the persecutions of Nero; he meets Jesus on the Appian Way: "And when he saw him, he said, 'Lord, whither goest thou?' And the Lord said unto him, 'I go into Rome to be crucified.' And Peter said to him, 'Lord, art thou being crucified again?' He said to him, 'Yes, Peter, I am being crucified again.' Peter came to himself, and having beheld the Lord ascending up into Heaven, he returned to Rome, rejoicing and glorifying the Lord, because he said, 'I am being crucified,' which was about to befall Peter" (James, The Apocryphal New Testament 333).
The Acts of Peter record the condemnation of Peter by the prefect Agrippa, his request to be crucified head downward, and a long sermon that he delivered on the symbolic meaning of the cross. This discourse betrays Gnostic influence, as do certain other passages of the Acts of Peter. About two-thirds of the text have been recovered; small Greek and Coptic fragments and the main body in a Latin manuscript were found at Vercelli (Actus Vercellenses ). Ambrose used the anecdote without reference to its Apocryphal character to show that, as Peter stood firmly with the Church, Ambrose would stand with the Church of Milan against the Arians. - Quo Vadis
Thus there is some reference to St. Peter in Rome from the late 2nd century, in the Acts of Peter, even though the authenticity has been questioned by some.
The earliest reference to Saint Peter's death is in a letter of Clement, bishop of Rome, to the Corinthians (1 Clement, a.k.a. Letter to the Corinthians, written c. 96 AD). The historian Eusebius, a contemporary of Constantine, wrote that Peter "came to Rome, and was crucified with his head downwards," attributing this information to the much earlier theologian Origen, who died c. 254 AD.5 St. Peter's martyrdom is traditionally depicted in religious iconography as crucifixion with his head pointed downward.
Peter's place and manner of death are also mentioned by Tertullian (c. 160–220) in Scorpiace,6 where the death is said to take place during the Christian persecutions by Nero. Tacitus (56–117) describes the persecution of Christians in his Annals, though he does not specifically mention Peter.7 "They were torn by dogs and perished, or were nailed to crosses, or were doomed to the flames and burnt." Furthermore, Tertullian says these events took place in the imperial gardens near the Circus of Nero. No other area would have been available for public persecutions after the Great Fire of Rome destroyed the Circus Maximus and most of the rest of the city in the year 64 AD.
This account is supported by other sources. In The Passion of Peter and Paul, dating to the fifth century, the crucifixion of Peter is recounted. While the stories themselves are apocryphal, they were based on earlier material, helpful for topographical reasons. It reads, "Holy men ... took down his body secretly and put it under the terebinth tree near the Naumachia, in the place which is called the Vatican."12 The place called Naumachia would be an artificial lake within the Circus of Nero where naval battles were reenacted for an audience. The place called Vatican was at the time a hill next to the complex and also next to the Tiber River, featuring a cemetery of both Christian and pagan tombs. - Saint Peter's tomb
It is no coincidence that Constantine built the first St. Peter's Basilica in 330, on Vatican Hill. Saint Peter's tomb is a site under St. Peter's Basilica that includes several graves and a structure said by Vatican authorities to have been built to memorialize the location of Saint Peter's grave. St. Peter's tomb is near the west end of a complex of mausoleums that date between about AD 130 and AD 300. The complex was partially torn down and filled with earth to provide a foundation for the building of the first St. Peter's Basilica during the reign of Constantine.
What might seem more than just a simple coincidence is that Our Lord said to St. Peter that he is the rock upon which He was to build His Church. When Pope Pius XI died (February 10, 1939), his successor Pope Pius XII ordered that a place be made available to put the body of his predecessor in the lower crypt area in the Vatican Basilica. In doing so workers discovered the ancient necropolis of Vatican Hill.
What they found during these excavations, which for many years was done in almost total secret, partly because of the World War II, was utterly amazing.
During excavations under St Peter’s Basilica that began after the Second World War, archaeologists discovered a funerary monument with a casket built in honour of Peter and an engraving in Greek that read "Petros eni", or "Peter is here". - Bones attributed to St Peter found by chance in 1,000-year-old church in Rome
Time Magazine has an excellent article with many photos of the excavations of the Vatican Necropolis: LIFE at the Vatican: Unearthing History Beneath St. Peter's
In 1950, LIFE reported on a years-long effort undertaken beneath the staggeringly ornate public realms of the Vatican, as teams of workers meticulously excavated the myriad tombs and other long-sealed, centuries-old chambers far underground. Nat Farbman's color and black and white images in this gallery — most of which never ran in LIFE — were touted on the cover of the March 27, 1950, issue of the magazine as "exclusive pictures" for the story titled "The Search for the Bones of St. Peter."
Deep in the earth below the great basilica of St. Peter's in Rome [LIFE wrote] the clink of pickaxes and the scrape of shovels in the hands of workmen have been echoing dimly for 10 years. In the utmost secrecy, they have penetrated into a pagan cemetery buried for 16 centuries. Architects feared they might disturb the foundations on which rests the world's largest church. But the workmen, with careful hands, pushed forward finally to the area where, according to a basic tenet of the Catholic Church, the bones of St. Peter were buried about A.D. 66.
The Church has always held that Peter was buried in a pagan cemetery on Vatican Hill. Now, for the first time, there is archaeological evidence to support this: the newly discovered tombs, which LIFE shows [in these exclusive pictures].
The greatest secret of all — whether the relics of the Chief Apostle himself were actually found — is one which the Vatican reserves for itself, although there have been rumors that the discovery of the relics will be announced at an appropriate time during the Holy Year.
NOTE: In December 1950 Pope Pius XII announced that bones discovered during the excavation could not conclusively be said to be Peter's. Two decades later, in 1968, Pope Paul VI announced that other bones unearthed beneath the basilica — discovered in a marble-lined repository, covered with a gold and purple cloth and belonging to a man around 5' 6" tall who had likely died between the ages of 65 and 70 — were, in the judgment of "the talented and prudent people" in charge of the dig, indeed St. Peter's.
YouTube has a few excellent clips of what lays beneath St. Peter's Basilica:
• Pope Pius XII and St. Peter's Tomb
• Spectacular Virtual Tour of the Tomb of St. Peter on Vatican Website
But do not take my word for it, skeptics are permitted to visit La Scavi. For those interested, visitors are permitted into the necropolis under St. Peter's Basilica. Restrictions may apply to certain individuals or groups. Those who suffer specific and serious physical problems that could be effected by these conditions, including claustrophobia, should not visit. It is worth visiting in Rome, especially when one can see the Graffiti Wall and the place were the bones of St. Peter were discovered. "Upon this Rock I will build my Church!" I have been there and would recommend it to all.
Special visits to the necropolis underneath the Basilica, where the tomb of St. Peter is located, are only possible following special permission granted from time to time by the “Fabbrica di San Pietro”. Visits are organized according to the schedule set by the Excavations Office.
