If you only read Acts 28, you might get the impression that Christian faith was fairly new to Rome at the time Paul arrived there. But in reality, Christians had been in Rome for a long time before Paul finally set foot in the city around AD62. The emperor Claudius had already kicked all the Christian Jews out of the city in AD49. Apparently, the controversy within the synagogues of the city had become so divisive that it even drew the emperor's attention. Priscilla and Aquila left Rome at that time (Acts 18:2).
We do not know exactly when Jewish believers in Jesus first came to Rome. Perhaps it was some of those present in Acts 2 on the Day of Pentecost. Still, it is hard to imagine the issue not coming to a head sooner if many Christian Jews had been in Rome for that long. By AD49, Christianity had apparently grown strong enough for all-out war to break out in the synagogues.
Christian tradition has long held that the church at Rome was founded by Peter, especially Roman Catholic and Orthodox tradition. But this suggestion seems unlikely. Peter probably did not come to Rome until after Paul had visited there. Acts gives us no reason to think that Peter had ever been to Rome before Paul, nor does the book of Romans give us any hints along these lines.
On the other hand, it is possible that the churches at Rome had the flavor of Peter’s Christianity. Is Rome the city to which the sermon called Hebrews was written? Is this the audience that was tempted to rely on the sacrificial system rather than on the death of Christ?
If so, Hebrews mentions a previous time of persecution. They had earlier endured a time of great conflict and suffering (Heb. 10:32). Some had been arrested (10:34). Some had lost their property. Perhaps some of their leaders had been martyred (13:7).
Some think that crisis might have taken place during Claudius’ persecution of Christian Jews. But this crisis sounds more like what happened around AD64 when Nero blamed Christians for the fire that burned much of Rome. That event was a couple years after the time Paul first reached Rome. Indeed, perhaps Nero’s encounter with Paul gave him the idea of blaming Christians for the fire of Rome.
After Claudius kicked the Christian Jews out of Rome in AD49, the Christian community left would have been overwhelmingly Gentile for a time. By the time Paul wrote Romans, not quite 10 years later in the late 50s, some Christian Jews presumably had returned to Rome. Romans gives us hints that the church at Rome at that time was still primarily Gentile (cf. Rom. 1:6; 11:13; 15:13). 
It is quite likely that we should think about the churches, plural, at Rome. They met in house churches that could scarcely hold more than 40 or 50 people in a very large one, and Rome probably did not have as many churches even this large because of tenement housing. It is thus quite possible that the church of Rome was more segmented than in some other places...
 The Roman historian Suetonius, writing in the early second century, mentions this expulsion in Claudius 25.4.
 Peter likely wrote 1 Peter from Rome. 1 Peter says it was written from Babylon (5:13), which probably means Rome (cf. Rev. 18:2). It is only natural that Jews would refer to Rome as Babylon after Rome destroyed Jerusalem in AD70 the way Babylon had. It is possible that they referred to Rome even before then, especially in the late 60s after the Jewish War had started. This would be necessary for Peter to be its author, since tradition has it he died at the hands of Nero. See my forthcoming volume, The Church Moving Forward for discussions of 1 Peter and Revelation.
 This is the hypothesis of Raymond Brown and John Meier in Antioch and Rome: New Testament Cradles of Catholic Christianity (Mahwah, NJ: Paulist, 1983).
 E.g., William Lane, Hebrews 1-8 (Waco: Word, 1991).
 Although it is common to hear that Hebrews was written to Jews, this is usually based upon certain stereotypes that don't hold up under closer examination. Hebrews 5:11-6:4 fits a group of Gentile converts to Christianity much better than it fits Jewish converts. The date of Hebrews is disputed, but if it was written after the destruction of Jerusalem, it could indicate that Christianity at Rome remained primarily Gentile later in the century as well.