For washing we need the backpacking equivalent of soap and towel.
Small pieces of ordinary soap can be used but they are messy and crumble, we use liquid soap which is sold in a pump dispenser and can easily be used to fill a small plastic bottle. The smallest practical bottle we found was a holiday-size hand lotion in Body Shop, which has a capacity of 30ml, see photo. The liquid soap is concentrated and this tiny bottle lasts a long time, 11 days was our longest trip and there was still some left. The weight including soap is 35g.
Our towel is a ZAP cloth: this is a thin piece of latex (I think) about 27cm by 8cm and has a strange cold rubbery feel but works surprisingly well. The ZAP must be kept slightly moist and comes supplied with a fairly thick sealable bag, but actually there is no need to fasten the seal provided it is moist at the start of the trip. After washing it just needs wringing out. The weight including the bag is 65g. There are other types of small light backpacking towel that work more like a conventional one but we have never tried them.
One should never wash directly in a stream or tarn, or even a small pool, as the soap pollutes the water and is detrimental to aquatic life. However it is not necessary to carry a separate container since the hands need not be immersed. We just fill a bottle and pour a small amount at a time onto the hands or feet, well away from the water source, this works fine and only requires a very small amount of water.
For teeth cleaning we use a small plastic beaker for the water and a travel-size tube of toothpaste (26g).
We use these small wipes for 'strict' hygiene applications such as before handling food in the evenings. They are also useful as part of the first aid kit for cleaning dirty cuts.
These are larger and thicker wipes with a higher moisture content and replace the soap and water. The various brands have different cleansing abilities, which is relevant in summer when we use suncream - we don't like the idea of getting SPF30 on the down sleeping bags, however small an amount. Quality suncreams are specifically formulated to resist removal by water and the cheap wipes are not up to the job. There is a simple way to distinguish the good ones - look for the statement "will remove waterproof mascara" - these should also be among the most expensive.
Physically similar to cleansers and provide the sweet smelling antibacterial finishing touch. The wipe system may not be quite as refreshing as normal washing but this combination of cleanser and deodorant works pretty well. The airtight bulk packs are rather heavy and weight can be saved by taking the exact number required in a small snap-top container.
There is an antibacterial cleaning gel that works like soap but without water. This can be found in outdoor shops but we have never tried it.
Advances in the antibacterial properties of base layers mean that from the point of view of hygiene and cleanliness, they don't need changing for a long time. We wore our Dryflo base layers for 11 days in mid summer with no detectable odour at all at the end, and we are sure they would have performed for considerably longer. However in the warmer months socks will only last 2 or 3 days before they speak for themselves, and the number of spare pairs depends on how much you can endure!.