This stage is hard! I probably died on this stage more than any of the others put together. Your own mileage may vary. Anyway. Grab the Heart Container and then jump onto the platform to start this stage off with a blast. Be careful and jump quickly and accurately because if you miss a platform, you're more than likely dead. You should also continually use your Spread Fireballs. You'll see why when you get far enough. So, when the platform starts falling back down, jump to the next series of platforms until you see one that's falling faster than the rest.
Jump onto that one and then quickly jump to the next platform on the far right. There will be three more quickly falling platforms, quickly jump onto them until you reach the more stable platforms above. Face left and kill the two Green Robots with your Spread Fireballs. Now jump onto the platforms previously occupied by the Green Robots and continue up until you see three more quickly falling platforms. Jump onto the middle one, wait until you have enough room to jump onto the one on the right, and then jump onto the stable platform above it. From there, jump onto the quickly falling platforms, and then to the stable platform on the left.
There will be two more Green Robots walking around on the two platforms above you. Jump and knock the Robots off each of the platforms, and then continue up onto the stable platform on the left. Once you make it, there will be another three quickly falling platforms. Jump onto them and then back to the left, where there's another stable platform. After this, there will be nothing but quickly falling platforms and the occasional rocket that will appear from the bottom of the screen. They appear randomly but if they get in your way, you'll be in big trouble. After climbing for awhile, you'll see a single stable platform above the rest. Jump onto it and you'll be taken to the boss.
This is the guy you see just before the title screen appears. Switch to your Explosion Fireball and wait for him to stop talking. When he runs towards you, let the fireball go to damage him. He'll continue toward you until he's close and then he'll lift his sword and try to hit you with it. When he does this, wait about a second or so after he lifts the sword, and then jump to avoid it. He'll keep going this until you hit him, so when he backs up and tries to hit you, charge up the Explosion Fireball and hit him when he comes at you again. This will make him jump to the far right, shoot out a stream of fire, and then charge you and try to hit you again. He'll keep repeating this pattern until you hit him enough times, and then he'll turn into a giant coin.