Depends on the Era. Under Constantine, there was actually very little persecution of Hellenes or any religions at all, despite all of the noise made about Constantine. Constantine was far more pragmatic than pop culture historians today, or even in the past, make him out to be. Towards the end of his reign, he was more opposed to the creation of new sites and did confiscate several properties, but this paled in comparison to future emperors.

Constantine himself was raised by a devoutly religious mother, who was the concubine of a Roman General/Statesman, Constantius. Constantius walked out on the family relatively early, but apparently they kept in contact, and Constantine was his only legitimate heir. Constantine, in his early life was apparently not very interested in Christianity, and seemed to have identified himself as part of the Hellenes, though he was by no means ignorant of Christianity. We know that he was in the army and took part in the various religious rites and practices of the time, and also was probably a devote of Sol Invictus. We know this because he wasn't purged from the military by Diocletian for refusing to sacrifice to the Gods. We see on several minted coins the usage of Sol Invictus' symbols. Sol Invictus was popular amongst the educated and enlightened elite as it was considered to be a more deistic, or at least monotheistic, cult, and was closely associated with Apollo.

At some point, this changed before he became Emperor in 306. There are several conflicting accounts, but when his army arrived outside Rome against Maxentius, they were, according to the sources, bearing unfamiliar symbols, which can be inferred to being either the Cross or the Chi-Rho Symbol (the first two letters of Christus in Greek). Despite being outnumbered, he won a crushing victory of Maxentius, who was a deeply unpopular leader in Rome.

During this time, Constantine upheld many of the pretexts of hellenism and the traditional Roman religion, relating to his clothing, personal insignia, and political statements, but privately snubbed several conventions. He disregarded auguries and failed to make the proper sacrifices at the Temple of Jupiter in Rome.

Additionally, Constantine did not make Christianity the state religion of the Roman Empire. That was Theodosius I, almost 100 years later. He did, however, grant major religious toleration to all groups, Jews included, but specifically outlined religious toleration for Christians and established it as his personal religion.