When God designed the cat, he designed a perfect creature. After all, can you think of another animal that's come with preprogrammed bathroom manners?
When you notice that the kitten is weeing and pooping on his own, you can start training him to use the litter box. Training is the wrong word. Showing him the litter box is more accurate.
First, don't use a clumping litter unless it's a corn or wheat litter. I prefer World's Best Cat Litter because kittens experiment with their mouths and taste everything. If he eats corn litter, no problem. He'll just poop it out. If you use a clay clumping litter, it could lodge in his intestines.
I will wee the kitten and leave the pee and poop in the box so he can smell it. I may also take some used litter from my adult's boxes, again so it smells like, well, a litter box. As soon as you feed him, place him in the litter box. He may immediately go to work or he may play--just like a kid. However, he doesn't get to leave the box until he's gone to the bathroom. When he wakes up, put him in the box. If you see him start to go, simply scoop him up and set him down in his box. Praise him. Never yell at him because he didn't make it in time. Sometimes when they're trying to poop, they'll cry out. Kind of a pep talk. Put them in the box. Praise, praise, praise. If he's out and about, put him in his litter. Activity seems to get things moving.
Don't expect him to be able to do his job if you're having problems with diarrhea. So get coccidia, worms and other intestinal problems taken care of.
Remember, lots of praise when he remembers--don't yell when he forgets.
I bet you moms out there wish kids were this easy!