You need a comma between "hand" and "cutting him." "Outburst" is one word.
Wow. Harry sure is mean in this one. That's not how he tends to act, even when lonely and frustrated.
"You're going to pay Snape."
This is Harry telling someone to give Snape money. Try putting a comma after "pay."
So... Harry goes from being super-meanie!Harry to being polite and calling Snape "sir"? That's more IC, but the transition needs work.
The exposition seems a little too good, the part when Harry explains to Madam Bones. I think he'd be a little more vague and hesitant.
"a lightning bolt with a serpent slithering up it."
Ooooooooooooo! Apt.
Snape strikes me as the guy who'd still hold a personal distaste for Harry regardless, and vice versa.
A unique and original take, but Harry seems awfully bipolar.