How did Christianity become the dominant religion of ancient Rome?
Through a series of Civil Wars and systematic oppression and extermination of dissenting views, the same way Christianity has become dominant anyplace else. There has never been a single Historical example of a nation being converted to Christianity peacefully without massive brutality and loss of life. It is the norm among polytheistic cultures to respect other Gods, they would, therefor, accept Christians without problem, the supposed “oppression of Christians” under Rome is largely a myth. The Christians, however, being monotheist, didn’t have the same sense of tolerance and would actively seek to undermine and destroy the people who had welcomed them in order to impose their own control over the society.