Welcome to Skull Castle, Dr. Wily's hideout.
The stage begins with you entering the castle through a large outdoor section. Pipis try to drop their eggs on you here, so whip out your Leaf Shield, stand still, and let the baby robot birds fly into it, killing them instantly. Collect the power-ups you get from them, repeating this strategy as much as necessary to avoid damage.
You'll eventually begin climbing the central shaft up to the main entrance. A number of enemies including Springers and Worm Dispensers will attempt to stop you along the way. Use Master Weapons where necessary, but save your Quick Boomerangs; you'll want them for the boss.
As the stage progresses, you'll come to a large cliff with a tiny ladder at the top. To reach this latter, you need Item 1. Make a platform, wait for it to rise for a few seconds, and make another to the left. Do this three or four times, and you should be able to make it up all right. Don't despair if you don't make it; it takes practice.
Climb the ladder quickly, as there are Terrys that will knock you off if you stop. Don't stop, and they won't get you.
Go into the Castle Entrance, and keep moving forward across the platforms and blocks. Eventually, the screen will begin scrolling and you'll have to keep moving to keep up. A massive robotic dragon will appear, making the trek all the more perilous. DO NOT touch the dragon - a single hit will kill you.
Eventually the chase will end and the Dragon Droid will have you cornered in a three-ledge section of the room. There are three blocks - top left, middle back, and bottom left. Stand on the top left, and unload on the dragon's head with your Quick Boomerangs. If he fires, jump back to the middle back block, alternating between the upper two blocks. If you fire the Boomerangs quick enough, this guy should fall quickly.