I strongly suspect that most backpackers/wild campers dont wash much at all whilst out in the wild bits. (Its just a theory but its uncanny, for instance that given a normal westerly flow of light breezes, businesses in the east of SCotland always seem to know when the TGO challengers are getting close.....)
But if you insist on not smelling like a yak herder's incontinent granny's favourite tights, you can get liquid soap from places like Millets - and its excellent for washing up with too. Keep the soap away from streams, though and pour the waste into small holes made in the ground specifically for the purpose using your special small hole making tool, or a tent peg.
The only really important thing is to have clean hands when handling food and cooking equipment - and as well as the soap, you can get alcohol-based gels (cuticura) from chemists. Its the same stuff as used in hospitals and is a handy accompaniment to a packet of tissues when away from your tent on a "mission"
You also can get quite a bit of benefit from soaking your feet (soapless) in a stream - the cold water seems to have a beneficial effect - or you can have a full dip if its warm enough and you dont mind some of your parts disappearing for a while (applies to males only)
If you need to wash clothes, you can do this quite effectively without soap by repetetive soaking , beating stuff against rocks and wringing/rinsing. It takes a while but it does work. The problem then arises about how to get the stuff dry again - this could potentially be impossible.
Whatever you do, dont use the alcohol gel on any part of your body which might be described as "delicate" - if you catch my drift.
Unless you like that kind of pain.