Each time a creature is put into a graveyard from the battlefield, check whether Blood Cultist had dealt any damage to it at any time during that turn. (This includes combat damage.) If so, Blood Cultist's second ability will trigger. It doesn't matter who controlled the creature or whose graveyard it was put into.
If Blood Cultist and a creature it dealt damage to are both put into a graveyard at the same time, Blood Cultist's ability will trigger, but it will do nothing when it resolves.