I've found little difficulty in playing the stages in this order:
Wood Man: Just evade the Leaf Shield when he tosses it at you. Normal shots will do enough damage to kill him rather easily.
Air Man:
Leaf Shield does wonders here.
Crash Man: Use the
Air Shooter because Crash Man likes being high to attack.
Metal Man: Might as well get him out of the way, because you need the
Metal Blade.
Bubble Man: Collect
Bubble Lead to use on Heat Man.
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.
Crash Man: Get his
Time Stopper prior to battling Quick Man.
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.