Why did Emperor Constantine begin to accept Christianity in the Roman Empire?
Most likely a combination of personal faith and political expediency
Constantine's mother was devote Christian. She was the daughter of an Inn keeper and a consort not a legal wife. Still as his father was a high ranking Roman official and out of the home frequently ,his mother would have had a large influence on Constantine as a young boy.
Constantine inherited control of 1/4 of the Roman Empire. This power sharing was unstable. Constantine marched his army of Germans and Franks to fight for control of the Western part of the Empire. Outnumber 4 to 1 Constantine told his troops that he had received a vision from God that they would conquer under the Christian the sign of the Cross. Their moral strengthened by the vision his troops vanquish his rival at the battle of Milvian Bridge.
The Dioclletian persecutions had attempted to extinguish Christianity in the Roman Empire. Christianity though declared illegal had continued to grow while belief in the ancient Roman gods waned. With the edict of Milan Constantine made Christianity a legal religion winning the support of a large part of the population. The support of the pagans population no longer could guarantee political power in the Roman Empire.
Later when Constantine had won control of the entire Roman Empire he moved his capital to the east where Christians were more populous and vocal. A t this time Constantine gave Christianity not only legal status but preferred status and the support of his government. Constantine's support of the Christian faith guaranteed the support of the Christian population for his government.
It seems that Constantine had a personal faith in Christianity as seen in his relationship with his Christian mother. Constantine also reaped political benefits by his acceptance of Christianity