Sorry, but I completely disagree with bathing goats unless it is absolutely necessary. It's pretty hard to wash them and have them completely dried and comfortable in the same afternoon. A wet goat is very succeptible to respiratory problems - wet goats get pneumonia. Young kids are even more at risk. Don't do it.
I don't even wash goats before show. Their pens are kept clean and they're bedded on pine shavings, so they don't really get dirty. Prior to show, they get the clippers and some basic cleanup (usually a waterless shampoo) if it's really needed. None of the other breeders I know wash their goats, either.
Chances are, your kid doesn't have fleas, but lice or mites. Even if the kid was with its mother, she wouldn't clean it like a cat or dog would. Lice can cause skin problems, anemia and underweight kids and needs to be treated.
Have you wormed the kid yet? Make sure you use a round of safeguard for goats (oral wormer). This is a pretty mild dewormer and you'll have to deworm twice. Follow instructions on the bottle. The added benefit of dewormer is it often treats external parasites as well.
For a topical treatment, start with a dusting of DE (diamotous earth). This is a completely safe and effective way to treat internal and external parasites. DE can actually be added to feed - it is powdered fossil that penetrates parasite eggs and eliminates them. Dust your kid with it but keep it out of the eyes and nose. You should see a pretty quick improvement.
If the problem is persistent, I use a couple of drops of Ivomec (for cattle, actually, they'd don't make it for goats) and give it to the kids orally. Ivomec is a very potent dewormer and I only use it when there is evidence of resistent parasites.
I had a fecal sample taken from one of my does last week and the vet tested it with no eggs (or worms) in my animals, so I can stand by my methods - no parasites, no lice or mites without baths or heavy duty chemical treatments.
ADGA and NPGA goat breeder