The fifth world features a lot more enemies which fire upon you, and many fire bars that are placed in locations that are difficult to avoid. By now, you should be much more comfortable leaping in the air in order to avoid attacks and taking large amounts of damage. Don't stand around picking a fight with minor enemies. If there's nothing worth searching for in a particular screen, keep moving.
What this world packs in firepower, it lacks in complexity. It is pretty much an oval with a few extra rooms at the top. From the start, move up and traverse north along the east side of the world. You will be able to collect a couple of Speed Potions if you need them, and an extra life granting Moai statue. To the left of the Moai room is the boss' hideout. But before you tackle it, be sure to head north as far as you can go to look for the Artificial Heart in this world.
The boss of World 5 is a mass of cells that is surrounded by four smaller masses that serve as shields. The smaller masses rotate around the center mass, and fire bullets throughout the room. Rocks are a good strategy to use against this boss, but it is far more effective to throw them up or down as opposed to left or right. Throwing them laterally increases the likelihood that the rocks will be blocked by the smaller masses.
When you are finished, collect the key and exit the room. Head to the left and find the warp zone leading back to World 4, but not before finding the Rock Case one room to the south.