Quirk: Knife!
Izuku Midoriya is unlucky, to say the least and life hasn't exactly been kind. Things weren't so bad in the beginning, but once he was diagnosed quirkless, well, it all went to shit. Everyone grew to hate him, even his best friend and own father.
Was it really that impossible to become a hero without a quirk? It doesn't really matter because to Izuku, the term hero is flexible. He'll be his own hero because he's learnt from his own experience, people are selfish and the only one he can rely on is himself.
Left alone to fend for himself in the world, Izuku goes out to change things on his own terms, despite his methods being slightly illegal. Keyword, slightly. It's only illegal if he gets caught, right?