Kittens sometimes seem to come pre-trained to use the litter box. At about 3 weeks old, just when I'm introducing the first taste of solid foods, I put out low sided litter boxes for them. In no time at all, they start using them for urination, and eventually defecation as well. There frequently are accidents at such a young age outside the box, but they get rarer as the kitten matures. By the time you get your kitten at 14+ weeks, she should have no problems using the box, as long as YOU keep it clean for her.
When you first get your kitten, make sure he knows where in his new home the litter box will be. If the house is large, it's better to have more than one box in different locations. Kitty may be in one part of the house, while the box is in another, and an accident happens because she's still small and can't hold it long enough to find the box. And remember never to scold kitty for soiling accidents - she won't understand why you are upset, and you could create "litterbox avoidance" issues.
As your kitty grows, accidents rarely happen. 2 major causes for soiling outside the box are 1) the litter itself or 2) illness of the cat.
1.Litter. There are many brands and types of litter. Some cats prefer one over the other. I raise my kittens to use a few different kinds, but not all. If the cat doesn't like the brand you are using, he will find someplace else to "go". He can develop a dislike for a litter he has used before. Luckily, there are many kinds available, and one will likely suit him fine. If you ran out of his favorite litter and had to buy the bargain brand at 7-Eleven, this could be the cause of his problem! And if you decide YOU want to switch brands - make sure you do this slowly. As in everything, cats don't like change. So lead them through it gently, mixing the old litter with the new until he gets used to it.
Also make sure the box is clean. A litter box should be "scooped" at least once a day to remove solid waste and clumps, if you use clumping litter. If you have more than one cat, scoop more often. And it's a good idea to have a litter box for every cat plus one. 2 boxes for one cat, 3 for 2 cats, etc. Give them some choices!
2.Illness. Cats can get urinary tract infections that cause painful urination. They can also get crystals that form in the kidney or bladder that can block urination. If your cat squats for a long time in the box and nothing happens, they strain painfully while urinating, or if there is any blood in the urine or it is an unusual color, or they are going outside the box routinely, see your veterinarian immediately. Urinary tract crystals can cause a male cat's urinary tract to completely block. This causes the urine to back up into the system, poison the cat and kill him within a short period.
3.Other causes. There can be behavioral causes for not using the litter box, although these are less common. Stressed-out people can cause stressed-out cats, and stress can be a big reason for not using the box. Is there a new person or pet in the household? Perhaps a pet that is scaring the cat or keeping her from the box somehow? Check the windows - is there a cat or other animal outside that she feels threatened by? Remember that cats are very territorial, and a neighbor cat roaming around in her back yard might cause problems. Urination is a cat's way of marking her territory, so pay close attention to exactly where the cat "goes".