Here are some excerpts from some study materials I have on this subject, you may find other sources, but this seems to be the most reliable I have found, There is also some information in the Antiquities of the Jews by Flavius Josephus, which you might wish to check out.
When Paul wrote his letter to Christians at Rome towards the end of his third missionary journey, he was communicating with what appears to be a firmly established collection of believers in that city.
Based on a study of relevant biblical and extra-biblical documents, it is generally agreed that non-apostolic Jewish Christians brought the faith of Christ to Rome in the early decades of the church.
After generating both interest and controversy within the synagogues, Christianity was forced to reorganize in the wake of Claudius’s edict against the Jews. The resulting Gentile-dominated church that received Paul’s letter in the late 50’s met in small groups around the city of Rome but maintained communication and held onto a common identity and mission.
apparently Peter was instrumental in the formation of the church in Rome and became the first Pope in that branch of the Orthodox Church which split in 1055 from the Orthodox church in Jerusalem.
Paul and Peter leave their mark on these believers, though they merely strengthen the work that had already begun to flourish in the capital city.
Beyond these main points, scholars still differ on the exact timeline of the birth and growth of the Christian community, as well as on to what degree Roman reactions against Jewish instability stem from disagreements about Christ.
When all is said though, the overall picture of the emergence of Christianity in Rome constitutes yet another significant example of God’s extraordinary work in the early church during the decades following Christ’s death and resurrection.