I recently (within the last few hours of play, I think) noticed that my character's eyes have turned red.

I have Hearthfire and Dawnguard... did I accidentally become a vampire? I've checked my active effects and have seen nothing talking about vampirism and I haven't done any Dawnguard quests. I have killed some vampires while just playing normally, though.

if they're orange ish and glowing you are a vampire. Otherwise.. no idea.

Are you sure you haven't been turned? Wait 5 days and see if you start to shows signs of a starving vampire.