How can I get rid of fleas on my kitten who is only 5 weeks old?

See if a product called "Revolution" is for sale in your area (usually from Vet's office only). It kills fleas in all stages of life, hence killing the fleas and their eggs at the same time. All it is is putting one drop of this tuff on your cat's back (between the shoulder blades) once a month and it will not only keep the fleas off her, but also protects against a bunch of worms/parasites. If you can't find this product, start with the flea baths & collars and good luck. Either way you must wash all your blankets, sheets, towels, etc, and vacuum your entire home to protect yourself from a flea epidemic. It sucks to do all that laundry but not as much as flea bites on your ankles.

At 5 weeks, the kittens are probably too young for most flea treatments. You are putting pretty strong chemicals on a young animal. I'd try the mechanical methods at this stage... flea comb and soapy water to drown the fleas in. You can wash the bedding, etc. to try and minimize reinfestation, and treat the kittens with commercial products when they are about 8 weeks old. I think... check the package for the recommended earliest age.

My apologies,
Froggy7 makes a very good point about the kitten's age, and the product I mentioned, Revolution, does state that it is for cats 8 weeks of age and older. I would follow Froggy's advice for now, and start with the meds when s(he) is of age. Stay on top of it or you will have a pesky problem on your hand.