Short answer: On the whole, with distaste for many aspects of their religion but with pride at their accomplishments.

Long answer: We have to look at the situation pre and post Constantine as a complex religious situation. There were many different cults that a Roman could belong to, ranging from secret societies to massive cult worship. The Christians were one of the most broad in this respect, containing something for everyone. Other big hitters were the traditional Greco-Roman pantheon, the various mystical-philosophical creeds such as Stoicism and Platonism as well as esoteric mystery cults like Mithraism and Isis worship. Regarding the Greco-Roman pantheon, they were viewed as purely demonic. Primitive Polytheism in it's most lurid, debased form. The stories of the infidelities of various Gods and the constant sacrifices needed to appease them were alien to the Christian understanding. However, this was often excused by reference to certain statements by famous Romans such as Cicero expressing skepticism as to the truth of the old myths. Many "pagan heroes" like Cato were followers of the great philosophers Plato and Aristotle, who tended to monotheism so they were regarded as proto-Christians. The secret societies bore most resemblance to early Christianity because they required much fewer accouterments than classical religion and they promised a positive afterlife for believers. They were thus seen as the most dangerous threat and accusations of secret society membership dogged many of the more lukewarm chrisitians such as the jurist Tribonian.

Also of note is the nature of the church following Constantine. It was no longer a sect of the most pious, it was all encompassing. Even those with no attraction to mercy and peace were expected to take part, which led to the church and state becoming co-mingled. The persecutions of Christians was not really seen as bad because persecution was bad, it was bad because it was the official religion and had a natural right to power. This can be seen in the quick turnabout between the church of the martyrs under Domitian to the Church under Gratian and his successors which banned all faith but the Christian faith under pain of death.