It sounds to me like you are asking for a brief history of the first 400 years of Christian history. Is that correct? Personally I'm reluctant to take the time to write this.

What actually took Christianity from being a despised minority to the official religion? When Constantine I came to power and issued an edict of toleration, estimates put the Christian population of the empire at 10%. Constantine's decision to embrace and favour Christianity greatly changed its social standing, and by the 350s estimates put it at a little over half the population. By the time of Theodosius I in 379 Christianity was the major religion and became the official religion.

While I don't think imperial backing was the only factor, I would say it was a decisive one. Cumulative status change and peer social identity would be another one. Active philosophical and evangelical work by Christians would be a third.