World 6 will be one of the bigger headaches in the game due to two reasons. The first is that there is an abundance of buildings that can be destroyed, which will make searching for hidden items quite a chore. The other is because this is the first world where you will may face the continual onslaught of long-range missile attacks. There are only two defenses against these missiles; stand and fight, or continue moving. Moving offers better chances, but it makes searching for items difficult.
If you warped in from World 4, you will actually start in the lower left corner of the world where the warp zone to World 8 can be found. From here, head north as far as you can go, and visit the upper left corner for the Artificial Heart. Assuming you just picked up the heart, there's little need to visit the Konami logo in order to summon Konamiman since you will already be at full health. On the way to the boss' hideout, be sure to stop and collect the Rock Case. Be sure to approach the hideout from any direction but north.
The boss of this world is a gigantic Slime. The residual goo that this Slime leaves behind as it traverses the room will gather up into small Slimes that will attack King Kong in order to protect the bigger Slime. One tried and true method of attacking this boss by this point is to throw plenty of rocks at it. The rocks that get blocked by the smaller Slimes will destroy those Slimes and clear the way for more rocks to hit the larger Slime. With fully powered up rocks, the Slime should go down in no time.
With the key of the sixth world collected, head east and north to the warp zone that leads to World 7.