Cats are very good about using their litter. They only will do their business somewhere else if they are sick or frightened. All you need to do is pop the cat in the box when you first bring him home, just to show him where it is.
If you use the same type of litter as the one he's living with now it will be a breeze. Also, the first you do when you get your kitten is to put it directly in the litterbox when you get home. Then the cat knows where he can find his bathroom and will venture back after he starts to explore the house.
I worried about the same thing when I got my first kitty. But everyone assured me that kittens just automatically know what to do, and that I shouldn't worry. And it turns out they were right. Puph was only 6 weeks old but already knew exactly what to do. I never had even one moment of trouble with that little guy.
I've had a few more kitties since then, and never had any problems with any of them except my most recent one (that's Chuckles, the kitty in my avatar pic). He knew what to do--but if I ever moved the litterbox (like to a different corner of the room, or to a different room), he would pee on the carpet in the spot where the litterbox had previously been. But I think in his previous home, he and his brothers & sisters just used a sandy spot on the yard, I don't think they had an actual litterbox. That probably confused him a bit but he's just fine now (and trying to help me type this reply--hehe!)