First, congratulations on the win! But to be honest, you should have lost. Your opponent played some weak moves allowing you to get at his king.
9. Kh2 is just awful. It would have been better to play 9. d4 undermining your strong point in the center. White may have feared the check 9... Nxh3+ but after 10.Kh2 where can your night go? Just back to f4, where White is going to trade it off. How many moves did it make? White's knight on f3 covers all of the dark squares around the white king so there is no hope of an immediate attack. With no pieces in play and no way of attacking the king, Black has no compensation for the piece.
27. Qf5+ forcing the trade of queens would leave Black with no way of continuing the attack and with a threat of a fork on e7.
What you should consider is 6.c3? Look at the big hole that is d3... How could you exploit it? White handed you a nice advantage there that you missed. I LOVE doing this to opponents. This game I am posting is still in progress so do not comment on the endgame, but if you look at my 16th move you will see a similar idea.
http://www.chess.com/echess/game.html?id=20873837
White never really gets his pieces developed and plays with two pieces down most of the game...
I would like to thank Chess.com, because I'm really learning a lot! A month and a half ago I started as a 1150 rated player. Now my rating is around the 1450! This is my best win so far. Did I win because of luck? Or didn't he play very well? Please comment :-)