In Castlevania it's summarized simply as, in that iteration of the story, that yes the Catholic church in all of them is indeed the "badguy" who typically starts shit with dracula. The Generic basics though is that the Monster hunting order was literally wiped out by dracula. The Belmonts, on the other hand, were family friends of dracula before he became dracula in the original and only killed dracula for killing belmont's bride to be.
In the reboots, dracula was a belmont who became a vampire after killing the lords of darkness and the family kept it a secret and kills vampires and monsters in a fictional world where they're labeled as heretics for acknowledging their existence.
