My first ever online chess checkmate: where could I have improved?

Earlier today, I won my first ever online chess game via checkmate, though I'm still a beginner (as if this being my first ever online chess win didn't tip you off) and I would like some advice on my play. I played black.

Your 13...Kf7? gives him a knight as you walk your own king into a pin. Why not the simple 13...Qd7?
Why did not you take back the knight 15...fxe4 as he has broken his own pin with 15 Bf4?
Why did not you take 17...Qxd4+ and win back the knight?
At the end you both had 12 minutes on your clocks, though this was a 15+15 time control. You both played much too fast.
Before you play a move think and make sure it is not a blunder.

Alright, that helps. Thank you!

of course, you got lucky that your opponent gave away a rook in a winning position. In the opening, 5...Ne5 looks bad to me. You're moving a piece twice instead of completing development and the resulting exchange gives you a (supposedly fixed) double pawn on black squares, just as your bishop. That's a permanent blight to your play. Your opponent didn't exploit that and went for opening lines instead (typical e4 player), that even worked...
At this point, I recommend you do some tactical training, because all these strategic issues count for nothing if pieces are blundered, mates are missed, etc.

Things that I think will help you most:
1) blundercheck
2) tactical training
3) not playing "automatic moves", resisting the urge of "playing the obvious move", more thinking
4) developing love for endgames

Link that I think will help you the most:
www.youtube.com/playlist?list=PLl9uuRYQ-6MBwqkmwT42l1fI7Z0bYuwwO
(if you already watched it, you can watch it again after some time)

I think you should first focus on tactical training for the middle game and basic endgames to avoid giving pieces and missing easy win. After that maybe you should consider improving your opening (you don't have to learn deep lines you can simply stick to the fundamentals)

With 15 minutes you could take more time in early game to think about the strategic outcome of your move :
-does it complete pieces development ?
-fight for the center ?
-achieve king safety ?
- enable active squares for the minor pieces ?
-does the pawn structure remain strong ? is there some weaknesses in both side I should focus on/exploit ?

For example 5).. Ne5 is a bad move as furunkel pointed out
11..f5?? is a mistake, the game is almost lost strategically as it breaks your pawn structure and your king's position becomes very unsafe :
12)exf6 Nxf6 13)Bg5 pins the knight on f6 while f file is open ; black won't be able to castle on time
12.. gxf6? 13)Nc3! : completing development and questioning the knight on e4, either trade on c3 or g5 leads to an easy win for white after Qh5+

Another strategic mistake is 10)c3
10)..dxc3 and black has an easy endgame after queen exchange as white's isolated pawns are easy targets
10)..d3 and white has trouble completing development while containing the d pawn

Tactics and endgames may be the priority but don't neglect basic strategy especially in opening. You'll see your games will become much more enjoyable as wins will rely more on your skill when it comes to coordinate pieces than just some random tactic or your opponent blundering.