Claire is a vampire; either she's always been one, or she gets turned at some point during the show, whichever you like. She's been surviving on bloodbanks and city animals, along with the occasional rapist who thought the pretty nurse walking home was an easy target. Thing is, that sort of diet's like junk food for vampires - it fills her up, but it's not healthy at all.
Matt finds out, and offers to feed her.
+10 for no established relationship
+ 100 for Matt doing it as repayment for all the times she's patched him up at first, before really getting into it
+ 1000 for D/s, with Claire as the Domme and Matt as the sub
+ everything for religious imagery