The Roman Empire most certainly did NOT become mostly Christian during Constantine's reign. While Constantine himself may have converted to Christianity and attributed his victory at the Milvian bridge to the Christian God, you have to realize that the Roman empire at this point was enormous and Christians were still a minority. Most people still followed Rome's traditional religion(or religions, seeing as many foreign religions and deities such as Mithras' cult had been adopted by this time), as well as being part of the imperial cult, traditions going back centuries. As you can imagine, it would be next to impossible to change this within the reign of a single emperor.
Rather, the change was, as most change tends to be, gradual. Following the rule of Emperor Diocletian from 284 to 305, the Emperor increasingly came to be seen as a larger-than-life, even divine figure, beyond the stature of mere humans. This was a far cry from Augustus' humble insistence that he was but one of the people, chosen to guide Rome by the Senate and more closely resembled the royal courts of the Hellenistic kingdoms and Eastern dynasties, with fantastic displays of wealth and power such as the custom of prostrating oneself before the emperor.
And one thing that tends to happen in such cases is that the social elites begin to emulate their ruler, in this case becoming Christian themselves over a long period. This conversion was, of course, aided by the "laws" enacted by Christian emperors: Constantine's "Edict of Milan" in 313 ensured religious tolerance and in 321 he officially designated Sunday "Lord's Day", a clear sign of his commitment to Christianity.
As the social elite became increasingly Christian, the traditional religion became more and more associated with the countryside, where there was no incentive or pressure to convert. Indeed, the word "paganism" is believed to have come from paganus: rural, rustic. If you're looking for a single event that ensured Christian dominance, it may be Emperor Theodosius' banning of pagan sacrifices, a cornerstone of paganism, in 391. However, that such a law managed to be enacted at all is indicative that Christianity was already widely accepted at this point.
That being said, the transition from paganism was hardly a one-way ride. Emperor Julian tried to establish polytheism as Rome's dominant religion in 361 but failed, allegedly because his beliefs (in line with Greek philosophical thinking) were too abstract for other to take seriously.
Sources: Most of what I've said comes from Thomas R. Martin's concise "Ancient Rome From Romulus to Justinian", Chapter 9 more specifically.
If my answer is too vague, then I apologize. I am no historian and I'm mostly just regurgitating what I've learned a tertiary source. Hopefully someone more capable can come along and answer your question in more detail.
There's no one defined cause of the growth of Christianity, there are multiple factors and it was never instant. I'm working away from my notes right now, so I can't give you my exact numbers unfortunately.
Urban growth is one. Many Christians were based in urban settings, because that's where social networks were. Though Christians may have been a minority to the entire population of the Empire, they made a majority of the urban population. That's why there are churches in the imperial centres like Rome, Antioch, Alexandria, etc. A Jewish diaspora in early Christianity was a significant factor in establishing social networks. Like any other place in ancient times, plagues were very common, especially in cities. Christians often cared for sick rather than retreating to the countryside, which built up immunity and made them look a little more powerful than the pagan Roman -though maybe it was just the services that attracted people. Kind of like a sense of community and belonging, Christianity appealed to people.
Families, especially the role of women, were pretty important too. Not only were they allowed some role in churches, but they were often the ones responsible for converting their husbands. Think about it, ideas spread first to the closest people to someone. The parents convert, then the children, then external family and even friends. Early "churches" were really prayer/mass/sessions held in private households, and usually held by rich widows. Women were also ones who helped out the sick, fulfilling a sort of maternal niche.
During and after Constantine's reign, after the religion became official, Christian hierarchy established itself gradually through the use of bishops in place of bureaucrats. More often, bishops came something like provincial mayors. Christianity gained even more legitimacy after this, and even after the fall of the western half, you can sometimes see the church officials holding places together.
Check out Stark's "The Rise of Christianity," it's history by a sociologist and a lot of models, but he covers all the factors in a very concise manner so that you're at least able to grasp what they're about.