How did Constantine’s conversion to Christianity affect the Christian Church?
Constantine's conversion in fact did little for the Church itself other than stop its persecution. It certainly did not turn it into a persecutor of other religions nor did he have anything to do with the creation of Church hierarchy.It did not become the official religion of Rome until the reign of Theodosius some years later.
Constantine himself was not baptised until he was on his deathbed, at which point he related the story of his vision at Milvian Bridge. There is substantial argument among historians that Christianity was growing increasingly prevalent, and that Constantine only issued the Edict of Milan which made it legal (but not "official") as a pragmatic move to prevent further dissension. It was further a pragmatic move on Constantine's part to call the Council of Nicaea to determine the nature of Christ. A vociferous debate had erupted which threatened to cause a split in the Church and possible dissension within the Empire. Constantine thus called the Conference to ward off any problems.
There is also substantial doubt that Constantine himself was anything near a Christian. Whatever his beliefs, they did not stop him from having his son murdered because he saw him as a threat to his own rule. Also, for almost all his life, he worshiped the "unconquerable Sun" god, known as Sol Invictus. It is entirely possible, even likely, that Constantine morphed Sol Invictus into the Christian God.
An excellent resource on Constantine and his influence on the Church is Constantine, Roman Emperor, Christian Victor, by Paul Stephenson.
Constantine's conversion to Christianity changed it from a persecuted religion to one that was in the position to persecute other religions. It also helped to make Christianity a much more formalized and hierarchical religion.
Before Constantine, Christians were persecuted from time to time under Roman law. After Constantine, this ended. In fact, Constantine and later emperors even banned various pagan practices, at least in law.
The Christian church also became more organized after Constantine. His conversion made the church more of an official body of the empire and that meant it needed more organization. This led to a clearer hierarchy and to things like set mechanisms for deciding between different interpretations of Christian doctrine.
In these ways, Constantine's conversion made Christianity a more dominant and organized religion than it previously had been.