Anytime an animal is sick, the best thing you can do is make an appointment at the veterinarian. Let a professional answer this and advise you. Why a kitten gets an upset stomach could be for so many different reasons that you need to rule out. It's best to ask a doctor.
Take him to the vet if he keeps vomiting. He may require fluids, and he may need to be on a special bland diet for awhile. Did you change his food or give him cow's milk? Either of those things can also cause an upset stomach.
i have always used chamomile tea for my kittens and the majority of the time it works well. it is cleansing and stimulates the bowels. often times this will work right away, if you see they remain ill it is important to take them to a vet.