I try to stay fairly clean on a long backpack. My strategies are several:
1. swimming: I swim in a lake once a day, preferably at camp for the night. I go in with my clothes on. It is unbelievably refreshing, and has to clean things off a bit.
2. wet wipes: I wipe my face and neck with a wet wipe before going to bed.
3. I use a small wash cloth, with a few drops of soap, for arm pits, face, crotch, arms and legs. Feet get washed in the stream. I don't care if the fish die downstream.
4. clothes: I wear one set and carry a spare set of t shirt, underpants, and socks. Day one, I start out with clean clothes. Day 2, I wear the backup set of clothes, and in the evening, I wash the first set. Every day thereafter, I wash a set of clothes (t shirt, socks, underwear).
to wash clothes, I use a sil nylon stuff sack, with a dry bag closure (it also has a shower head). I put in a gallon or less of water, a few drops of soap, the dirty clothes, and I trap a bunch of air in the stuff sack, and seal it off. Then I shake it left and right, dump the water. Then put in rinse water, shake shake shake. lay them on a rock in the sun, wear the underwear to bed even if they are damp, if the other set is dirty.
I use the syl nylon stuff sack for other things, like for food or clothing.