Having a cape, feather and Yoshi make any level easier and this is no different. If you do not have a cape use the 2 turtle shells to kill the football-throwing Chucks. Use another shell on the third Chuck. You can also spin the cape to kill the Chucks.
The green pipe (found about halfway through the level) will lead to a surface area consisting of four platforms and a Hammer Brother. The platforms will fall once you land on them, so you need to move quickly. There is nothing of interest in the secret area.
Using the cape, float up to the ledge before the checkpoint and get the checkpoint before exiting through the pipe to the next area.
Throw a koopa shell down hill ahead of you to kill all the enemies. If you can keep up with it you can get an extra life from all the kills.
As you fall down the slopes, beware of regenerated or regenerating koopas. Also, remember to jump over the black clams of death, where the second row of coins are, when falling to the right.
You will eventually arrive at the pipe to Part 3.
Having a cape here is also extremely helpful. This screen is difficult, speed is vital.
Navigate up the "stairs" and over the red spiny turtle. You will find numerous ledges floating in lava pits. As soon as you touch the first yellow ledge it will start to sink, jump to safety (this one is easy). Again navigate the red spiny turtles. Then there are a series of flat yellow ledges. When you touch the first one, they all start sinking, so you must sprint across them very quickly.
Carefully jump over or spin cape the Chucks, if you do not have a feather in reserve, get the Fire Flower from the box instead, and proceed to the next series of slanted yellow ledges. It's still best to have a cape. Sprint across these ledges as well and jump and float to safety near the end.
The arrow sign is in sight. Go through the pipe, and the Exit is on the other side.
Cross the finish line.
Back on the world map, the green tunnel after Chocolate Secret brings you to the surface, bypassing Chocolate Islands 3 & 4 and bringing you next to Castle 6.