Constantine's conversion was a gradual process and there were most likely both personal/genuine reasons to convert in addition to more cynical motives, such as to appeal to the Christian minority within the empire. It really depends on how you view the evidence. Also, Constantine's opinion can and did change significantly - in his later life, when he was giving speeches such as the Oration to the Saints and adding Christian symbols to imperial paraphernalia, there is no reason to see him as anything other than a devout Christian, even if he converted originally for less than pure reasons.

Personally, I see the conversion as a genuine change of heart by Constantine. After all, before his victory near Rome Constantine already had Christian advisors and relatives at his court, such as his mother Helena. His father similarly had protected Christians in his part of the empire, whilst other emperors like Maxentius were happy to grant Christians the right to practice their religion (possibly only to annoy his enemies in the east, but it was still a degree of toleration at a time of persecution). Though the Later Roman Empire was autocratic, junior emperors were often happy to ignore the 'official' policy of the senior emperor and treat Christians leniently.

Even persecutions, such as the Great Persecution of Diocletian, are evidence of this, since they were not continuous nor were they always implemented with the same severity. They only occurred occasionally and under certain emperors. Lactantius, the Christian writer who later lionised Constantine, was for instance first invited to the imperial court by Diocletian to teach rhetoric - he resigned when Diocletian turned against Christianity, but clearly Christians were not excluded from the political world at this point, instead they were thoroughly integrated into it. Timothy Barnes recently also suggested that the Great Persecution was an attempt by the Caesar Galerius to influence imperial succession, as he knew that Constantine and Maxentius were Christian sympathisers, so by persuading Diocletian to do this, it effectively eliminated the two as credible successors to the Tetrarchy. Even before Constantine, Christianity was already a tool for those in power in addition to being the religion of many Romans!

Constantine's choice of Christianity was also a small step from the traditions of previous emperors. Paul Stephenson for example listed many emperors before Constantine who had gradually drifted towards a form of monotheism, from the Severans all the way to Aurelian, most of them with a particular focus on Apollo/Unconquered Sun. It was thus I think a short leap of imagination for Constantine to go from believing that he was protected by a divine patron to believing that it was the Christian God protecting him, especially as he was always in contact with Christians.

We can for example trace how his own ideas changed in the critical years before Milvian Bridge. This is because Constantine's famous vision actually occurred in 310 (if we agree with the recent analysis by Barnes), two years before the battle. An anonymous panegyric produced in Gaul in 310 referred to a celestial phenomenon seen by his army (now considered by some historians to be a solar halo) and even the church historian Eusebius placed this before the invasion of Italy in his Life of Constantine, while his earlier Ecclesiastical History didn't mention this sign at all. There was allegedly a dream during the Italian campaign according to Lactantius, but Constantine was told by God to put the 'sign of God seen in the sky' on his soldiers' shields, rather than something he saw in the dream, so evidently he already had an idea of what it should be.

Recent scholarship (P. Stephenson's Constantine: Unconquered Emperor, Christian Victor and T. Barnes' Constantine. Dynasty, Religion and Power in the Later Roman Empire in particular) also argued that Constantine and his soldiers first interpreted this as a sign of divine protection from the god Apollo, as Constantine was marching against Maximian in Gaul at the time. Only later did Constantine see this as a sign from the Christian God, which may have led to the six-pointed symbol of the sun being altered to form a Chi Ro instead. Constantine was tolerant and perhaps even a sympathiser of Christianity before 312, so his 'conversion' was nowhere near as dramatic as Eusebius and Lactantius made it out to be and was the result of gradual evolution. This isn't to say that Constantine's conversion was entirely pragmatic, but only that Constantine was living in a context where it was common to worship one god, so his conversion was not a break with his Roman context, but one very much in tune with it, as we can see from how his own interpretation of his life changed.