The major starting factor to the spread of Christianity was Constantine. He made Christianity the religion of Rome and built Constantinople, the new Christian capitol of the eastern Roman Empire.
Christianity did not have very easy though. They were widely persecuted, and most everyone had contempt for them. Because of this, martyrdom and massacres abounded.
But even through all this trouble, Christianity made progress in Rome. Constantine signed the Edict of Milan in 313 after assuming his position in 306 AD, and Christianity thrives under persecution.
The story of Constantine's conversion is a strange and somewhat famous one. In this story, Constantine and his troops see a bright cross with the inscription: "In hoc signo vinces" which means: "In this sign you will conquer". He also has a vision in which he sees Christ, who tells him to march against his enemies.
From here, with the help of a good jump-start, Christianity is able to spread far among the different colonies and empires. Of course, later there are evil emperors who seriously try to impede the Christians, but they never completely die away.