This is fairly typical behavior for a young kitten. The best way I have found is to start an interactive play time with the kitten, one on one, about 15 minutes a day where I get down to her level and let her chase things if she is so inclined. Another thing to do is not use cat litter for awhile, instead use either strips of newspaper, or even rabbit pellets. Makes a lot less mess to clean up. Offer her a second box, a clean one with a soft blanket inside that she can lie in, or get her a playmate if possible, a second kitten to help her with her activity level. Buy a large door mat or piece of astroturf and put this under the box with enough left over to catch all the litter she is spreading about.
Having watched this type of behavior on the new kitties that arrive here all the time, I think they just like the noise the litter makes and have not been with the momcat long enough for her to show them litterbox manners.
Good luck with your new kitty! What is her name?
Also I will be moving this thread to the Behavior forum for you. Lots to new kitty owners there to help you!