Two things.
1) Do not, under any circumstance, be above Bowser.
But more importantly,
2) DO NOT, UNDER ANY CIRCUMSTANCE, BE ABOVE BOWSER.
I like using Bowser. He's got a couple of really nice combos and kills at very low percents. It's all about getting in that first hit though, because he's pretty much combo fodder himself. If they land a hit on you that wasn't armor'd, you've lost the stock. This is probably why I (ab)use his Downsmash liberally, the SA frames come out pretty quick.
When fighting him, it's all about Spacing. Know his reach and stay out of it with Marth, Ike, or Link. Know when to dodge and when to punish, because his moves are stupidly punishable. Back when I first started trying out everything in 2.6, Bowser's AI gave me a hella hard time to the point where it took real work to win. After learning how he works, though - well, he's pretty predictable. And once again my go-to level 9 punching bag when trying out a new character, heh.
Edit: Also, his size, man. On the one hand I like it because it gives him stupid huge hitboxes and there's something to be said about it on the casual level. The sheer amount of space he controls in a 4v4 makes him terrifying. At the same time, on a competitive or even just a casual 1v1 level, he's a huge hurtbox asking for percent. He's still one of the best designed characters in the game though - the PMBR did a wonderful job buffing him up in a way that seems natural to his playstyle.