The persecutions of Christians started in Rome in 64 AD. Years earlier, in 49, Jews and Jewish Christians had be expelled from the city because their monotheism and their in-fighting was seen as civil unrest and disturbing to the Pax Romana. By 54 they must have eventually drifted back into town and taken on a lower profile but by then Gentile Christians probably outnumbered Jewish counter-parts.
We can guess that their community existed before Paul's letter because it is a letter of introduction to them and a detailed laying out of his theology. It appears to be a community strong by the 40's, within years of Jesus's death and Resurrection. Initially it appears to have been a Jewish Christian community, maybe even founded by those inspired by Peter's speech at Pentecost. This can be conjectured from the expulsion order in 49. There is no evidence of apostolic visits before that.
But by the 60's the community was more diverse, as Paul's letter indicates. Still it was strong enough community to warrant a letter and the request of support for further missions to the west. Solid enough then for visits not just from Paul but from Peter as well.
It says something also to the Faith of this community that Paul was able to wait to turn to it until after feeling that he had established the Faith securely in the East. The 20 years of strong community before his letter possibly meant that he could put off such a journey. It also meant that it was strong enough to serve as a base of operations in the West.
Not that there were not problems, as the letter tells us, but the letter also indicates that they were familiar with and mature enough in Faith to handle the depth, breath, and length of the letter and its theology.
This level and strength of Faith are borne out in Nero's persecutions starting in 64, and involving both Peter and Paul; how they must have witnessed to each other! How they witness to us now!
We are not a new philosophy but a divine revelation. That's why you can't just exterminate us; the more you kill the more we are. The blood of the martyrs is the seed of the church. You praise those who endured pain and death - so long as they aren't Christians! Your cruelties merely prove our innocence of the crimes you charge against us. When you chose recently to hand a Christian girl over to a brothel-keeper rather than to the lions, you showed you knew we counted chastity dearer than life. And you frustrate your purpose. Because those who see us die, wonder why we do, for we die like the men you revere, not like slaves or criminals. And when they find out, they join us.
-- Tertullian, From Apologeticum