This is a very difficult question to answer because any type of reasoning is mostly speculation. To begin with understand that by the time Constantine the Great came to power in 313 and begin the easing towards and eventual acceptance of Christianity it had been almost 300 years since Jesus death and the fact that the Romans had killed him barely even created a blip. Also understand that the Gospels of Matthew, Mark, Luke, and John all place far greater blame on the Jews for having Jesus killed. Pilate, for his part, is shown in an almost sympathetic light of being in an impossible situation and almost forced by the Jews to have Jesus killed. Also important to note is that Jesus death was all part of the plan according to the Bible, far from hating Pilate or the Romans they are all seen as merely playing their part in God's plan. Thus the fact that the Romans had killed Jesus likely didn't bother Christians and forgiveness is also are large part of the New Testament.

Now that we have that out of the way we reach into very murky waters of why the Roman government gradually shifted and eventually having Christianity replace their traditional religion as the dominant religious force of the Empire. To begin with it all starts with Constantine the great's "vision" in 312 shortly before The Battle of Milvian Bridge. Now, supposedly he looked up at the sun shortly before the battle and saw a cross of light accompanied by the latin phrase "in hoc signo vinces" translated as "in this sign, you will conquer". Note of course that the sources for this vision and decision by Constantine to have his troops paint the Chio Rho on their shields before battle are christian sources, so take them with a grain of salt. Regardless, while impossible to determine whether he truly had a vision, Constantine definitely spurned tradition by failing to offer sacrifices at the various Roman temples upon entering Rome and instead proceeded directly to the Imperial Palace. This break from tradition would become the first in a series of moves by Constantine throughout his reign that would shift Christianity from religious exiles to religious dominance. The culminating victory and arguably his most important achievement was the Edict of Milan which granted religious tolerance towards Christians and returned confiscated Church property. Constantine would go on to become a large patron of the Church, however he didn't receive baptism until his deathbed and continued certain practices that Christians frowned on until his death, and his supposed conversion is a source of great historical and religious debate. Suffice to say that it is a mystery that will go unsolved and his personal beliefs aside he undoubtedly provided the foundation of the Christian Church's dominance that would follow his reign and has continued to this day.

Yet, the more compelling question, and the reason for so much controversy and debate is why Constantine chose to adopt Christianity for the Roman Empire. Christian scholars will naturally argue it was a true conversion and religious experience and tout it as evidence of the divine power of God. Secular scholars will contend that he did so with an eye on the politics and expediency. The truth, like most things, likely lies closer to the middle than the extreme. It is impossible to know which of those is true and certainly visions in the sky seem improbable, though not impossible for a man who was by all accounts entirely sane. On the other hand Constantine hardly required the backing of Christians to claim power in Rome or for that matter the Empire and while they were a growing minority they hardly represented a significant political or military obstacle to his power. Thus we are given a conundrum and are forced to decipher through the lines. As previously stated Christians were a growing minority and perhaps Constantine decided to generate goodwill with a growing base from which he could generate almost fanatical support without firmly committing to them for the time being. Perhaps he wanted to believe he was God's handpicked man a fact which would likely have played to any man's ego. Or perhaps he truly was a man of devout faith who genuinely converted and carried out the will of God. The simple fact remains that for the time being we'll likely never be able to divine his true intent or belief and instead we'll continue to devolve into this patchwork speculation on a man more complicated than most by the requirements of faith, politics, and warfare.