My new 7 weeks old kitten will only use the litter box a few times during the day. The other times she will use random corners through out the room. We have 2 litter boxes she has found and I clean it out twice a day. She seems to find her way around my apartment ok, just not when she has to potty.
Kittens don't know enough to plan ahead and they will just play, play, play and then when the urge to go hits, they can't always remember their way back to the right place to go potty. I had this problem with my Bailey. Her tiny little legs couldn't make it all the way back up the stairs and she started using the corner of the living room. I had to put a litterbox there temporarily until she got bigger. I didn't especially WANT a litterbox in the living room but the alternative was pee on my rug. Also make sure the litterbox has low sides and is easy for a kitten to get into and out of. Another thing I did was to pick her up and take her back to the litterbox at least once an hour to help remind her to go in the box. If you put the kitten in and scratch around in the litter, they will often go potty for you even though they didn't think they had to go before.
"One cat just leads to another" - Ernest Hemingway
I did have the same problem when I first brought my kitties home, however 7 weeks of age still too young for a kitty I'm surprise they let her go so soon, but any way there is a very good cat litter trainer is made by doctor Kelsy, you can find it at any pet store, It's extremelly good, my kitty was using the bathtub as her toiler, but I soon I have change her on the new litter by doctor kelsey she inmediatelly start using it until this day, she's now 2 years old.
When our kittens are old enough to start using the litter box we use something shallow (like a cardboard box that you buy cases of pop in) and put a litter bag liner over the box...for litter there is a litter made just for kittens that has a litter box attractant in it.It is in a pink and blue bag and says Kitten Atract on the bag. It works great and makes litter training very easy! There is even a money back guarantee on it...here is the PetSmart link http://www.petsmart.com/product/index.jsp?productId=2754846 Good luck and happy Holidays!