I've found little difficulty in playing the stages in this order:
: Just evade the Leaf Shield when he tosses it at you. Normal shots will do enough damage to kill him rather easily.
: Leaf Shield
does wonders here.
: Use the Air Shooter
because Crash Man likes being high to attack.
: Might as well get him out of the way, because you need the Metal Blade
: Collect Bubble Lead
to use on Heat Man.
: I recommend doing Heat Man after killing Air Man so you have the Jet Sled
to cross the large pits found in the level. Remember to recharge it after crossing.
: Get his Time Stopper
prior to battling Quick Man.
: Time Stopper stops him in his tracks and drops his life by 1/2.
Weapons to use against the bosses of Dr. Wily's Castle.
Dragon: Quick Boomerangs do wonders here.
Block Robots: Use the Metal Blade to destroy the robots once they come out of the walls.
Guts Dozer: Not fast enough or strong enough to take a beating with Quick Boomerangs.
Homing Seekers: Immune to everything except Crash Bombs. Enter the battle with a full charge.
Teleporter Room: Fight everyone all over again. For Metal Man, use the Metal blade. Kills instantly! Other writeups in this node have suggested weapons to use. Keep a full charge for the Bubble Lead, because you'll need it in the final battle. In most cases, and against Wily, you'll be using the Metal Blade.
Alien Dr. Wily: Only thing that will harm him is the Bubble Lead.