Paladin (Vengeance) with Urchin background. He's out to find (& slaughter) the people who killed his parents.
This one doesn't feel too odd to me at all, but Warlock with Sage background. He was flunking out of the Academy, found a forgotten tome in the library one night that seemed to detail a shortcut to the magical talent he lacked (lowish Int, certainly for someone trying to make it in wizarding school), & attempted the ritual. Wound up bound to Something Not Very Nice - oopsie! - but now he's hooked on the power it gives him.
Also a dwarven sorc w/ Acolyte background. Dutifully following in the family footsteps to be a cleric, but his magic came out differently. That was when he learned about the big family secret - red dragon way back in the lineage, oh the shame!
One of my players is a cleric/charlatan. We worked out a story where he was a swindler, essentially peddling Pelor flavored snake oil, when the gods showed some cruel irony and granted him the ability to perform miracles. Nothing has gone his way since.