After toilet: if enough water available, I'll take a sip from my water bladder and spit it over my fingers. If no water, sometimes sanitiser, very occasionally an antibacterial wipe - usually I save these for the evenings so they are utilised more fully.
Wash dishes - hot water and a scourer/sponge which I cut into smaller pieces. River sand is also good for scrubbing but requires more water to rinse.
Before cooking/eating: similar to above.
Can highly recommend the body & hair products from these guys:
https://www.beebeauty.com.au/ Their sample packs contain good sized soaps/shampoo bars for bushwalking.
They don't make anything specifically for washing dishes or clothes though.