Introducing man-made substances to the waters of a wilderness is a big no-no, not only for the Leave No Trace principles but also just for common sense. If you bathe in a natural water source and use soap, any wild animal that drinks from that water source and all other sources it flows into will be drinking your soap. Swimming in and bathing in wilderness lakes and other large bodies of water are allowed if you skip soap. Smaller bodies of water can be contaminated by your body’s oils, so stick to large pools if you bathe at all. Not bathing is an option–in fact, ideally you shouldn’t bathe while out in the wilderness unless you really feel that you need to.
There are two methods of bathing in the wilderness: bathing without soap in a large body of water, or bathing by carrying water at least 200 feet away from the lake or stream where you got it and washing with only phospate-free, biodegradable soap. If you choose to bathe in the water, you can still get clean without soap. Pebbles, sand and gravel are useful to scrub with—think of them as natural loofahs. Bathing in the water is a handy time to wash your clothes, as well–swim in them and scrub them at the same time.
If bathing with soap, you must carry the water far away from the source. Bathe downhill from the source so that it won’t stream soap back into its original pool, and when you rinse your body and hair, try to spread the water out so that it doesn’t all land in one spot. The same goes for brushing your teeth–use salt or baking powder in place of toothpaste (which is a man-made substance), and try not to spit it out all in one spot.
Washing your hands, a vital part of everyday hygiene, is a bit trickier in the wilderness but still doable. You have the choices of either carrying water 200 feet away from the source and washing with soap, or using alcohol-based hand cleaner, which requires no rinsing to sterilize.