The first part of the fight has you fighting a flying Colonel Red, it should be easy enough to keep avoiding his bombs while hitting him and he doesn't have too much health.
Next comes Orange. It's quite hard to avoid taking damage on this boss, one thing to always keep in mind is to avoid the edges of the platform as falling or being knocked off will cost you 15 health. His first attack sees an energy wave being sent across the ship which you can swing down below the platform to avoid, you can't jump over it (don't hit jump twice when swinging down or you'll jump off and take 15 damage). He will also try and dash across the platform on occasion which you can also jump or swing below to avoid if timed right (this is the one that usually hits me though). When he jumps up and catches the rotor just keep on the move and his elbow drop should miss you when he jumps off (a slide will also help avoid it). If he moves over to you when below the platform he'll perform several quick attacks down, easiest way to avoid this is to hit him for enough damage just before so he goes invulnerable (and doesn't attack) for a short time.
If he gets too close to you he'll perform a damaging throw so try and avoid that at all costs, however most folk don't know that if you're quick you can perform one of your own and is worth a shot if it becomes unavoidable that you'll bump into him.