Backstory to Romans
Paul wrote Romans from Corinth, probably during the winter of 57-58 (see Acts 201-3)
As the capital of the Empire, Rome is the most important city in the Mediterranean world, and Paul had long wanted to visit. Now, at last, he believes the time has come for him to go to Rome. So as he's preparing to wrap up his third missionary journey and visit Jerusalem (bringing an offering to the Jewish christians there), he writes to the churches in Rome to ground them in the gospel of God's righteousness and to prepare them for his visit.
Rome was a massive city, boasting a population of over a million citizens which did not include all the slaves - and some estimates put more slaves in the city than citizens. So who knows how many people actually lived there. But it was huge. Here's a model of the city built in the 1940's.
The church in Rome began and grew without any direct connection to an apostle. Because of that, Paul felt free to include them in the sphere of his ministry and he wanted them to serve as a launching point for gospel ministry further to the west, all the way to Spain if possible.
To learn more about the background to Romans, watch the two-part intro video at the Bible Project: