What is the best way to get rid of fleas on a kitten that is too young to use store bought flea medicine?
As well as James' suggestion for Diatomaceous earth, I'd recommend using a flea comb to get fleas off the kitten. It's a nasty job (I've done it for cats who were too old to safely use commercial treatments) but it does help.
You sit yourself and the kitten on a plain light-colored cloth so you can easily see any fleas. Comb the kitten and use your fingernails to kill any fleas the comb picks up. You'll need fingernails because fleas are tough (it's rather gross - when you kill them you'll hear a distinct pop and there'll be white ooze. If the flea has been feeding, there'll also be blood).
Once you're done - usually when the kitten is too restless to keep going (with a kitten, you want to do this when its sleepy), put the kitten inside and shake the cloth out outside. That gets the worst of the flea dirt and flea larvae off. Then wash the cloth in the hottest water you can use, and use a dryer to dry it. The heat might kill any remaining fleas (this isn't a guarantee).
If you use tissues instead of cloth to catch the flea dirt and so forth, put them in a trash can outside the house.
Depending on how badly the kitten is infested, you may want to do this several times a day - kittens and elderly cats can suffer from flea-infestation anemia.
A bath with dawn dish soap is the safest and most effective method in my opinion. The lather from the soap drowns the fleas, and contains no medicines that can have side effects. In fact, I would say it's the most effective treatment aside from preventative medicines.
As with normal baths, just make sure to be careful not to get water in your kitten's ears, and just lather up the soap for a few minutes. You should start to see the fleas start to wash away. Follow up with a flea comb every day, and another bath when you notice any more fleas showing up after eggs have hatched.
Diatomaceous earth is best for killing insects in carpet and upholstery I think. Applying it directly to skin can cause irritation and dry it out if you apply too much. If you use it, I would suggest a light sprinkling while avoiding the face (to avoid irritation in the eyes and nose).
I really don't suggest diatomaceous earth.
My gut feeling is that it is more damaging to larger more complicated animals (like cats, humans,...) than it is to small ones like insects. Insects have hard exoskeletons that are more difficult to penetrate / damage than skin. Likewise, mammals have lungs (true insects don't) that are also easy to damage.
Diatomaceous earth is a very 19th century way of treating things, much like using borax for everything. My main objection is that they are probably less damaging to the insect than the host animal.
I have no idea how effective soap is on fleas, but I have an easier time believing it could do some good. Most insects get air through pores in their exoskeleton. Soap can clog those pores.