Clean the closet out and vacuum it thoroughly. Don't forget to vacuum all the cracks, crevices, and ductwork. Clothes moths are actually the same species that infests dead animals in order to eat their fur. This is one of the many reasons that poisoning rodents is a bad idea. A dead rodent in your walls is an excellent breeding ground for clothes moths. It is much better to use snap traps so you can dispose of the bodies!
If you have clothing or blankets that are infested, you can kill off the little buggers by sealing the infested articles in black plastic garbage bags and setting them in the sun to roast. (In the winter, you can put the infested articles outside to freeze, or, if you have access to a really big freezer, you can freeze infested items in the summertime.)
Cedar oil and lavender oil both repel moths. In order to prevent a re-infestation, be sure to clean woolens and silks before storing them. Clothes moths are attracted to food stains, sweat, hair, oil, urine, and starch. Completely clean fabrics do not contain the vitamins and minerals that the moths need to thrive. If you own any furs, they should be stored in cold storage only, because they are really much too attractive to moths!
Fish oil trap: unwind some flypaper onto a piece of cardboard and put some tiny drops of sardine on the sticky surface. The moths will be attracted to the fish smell, and will stick to the flypaper.