Rome and its empire were hugely important for Christianity, as Christians were raised in a Christian/largely Christian empire and were firmly ruling it from the late fourth century onwards. The same is true for the Romans - they were both Christian and Roman, and most people saw nothing contradictory between the two. Of course, some people took their faith seriously and others didn't, and emperors often tried to impose their viewpoints on the Church, but it is also clear that some bishops were fiercely independent and were willing to disregard imperial orders. There really isn't any generalisations we can't make about how the Church and state interacted, especially as orthodoxy often changed when different factions were dominant, so I don't think there can be a firm answer on such a broad question - if you have any more specific queries on Christianity after Constantine, feel free to ask them!