The beginning of the Roman Empire as a Christian empire lies in 313, with the Edict of Milan. The edict was signed under the reigns of Constantine and Licinius. The edict established tolerance for Christianity throughout the Empire, but did not yet make it the official state religion. After the Edict was proclaimed, however, the Christian Church rapidly became extremely influential amongst the ruling classes of the Empire, and the Bishops were established in positions of power and influence.
Christianity became the single official religion of Rome under Theodosius I (r. 379–395). The emperor had a considerable degree of control over the church. While Christianity flourished, the Empire by no means became uniformly Christian; paganism [Mithraism] remained significant. Theodosius massacred Thessalonica for rebelling against his new Christian policies condemning homosexuality, which was a common practice in both ancient Greece and Greece under Roman rule. Upon his return to Rome the Bishop Ambrose refused to let Theodosius enter the church until he made a public repentance. Theodosius did so, and from then on the church's powers grew. Eventually the church would gain enough power that it would outlast the empire in the west.
It's good to know your history. LINK: Roman Empire