Now this will seem silly to anyone who's not been camping and not actually experienced a small amount of water for use in cleaning. Now we have lots of water, almost a seemingly endless supply at the tap of not only cold but hot water. These were an incredible technological advance by any civilization in history.
While I've offered many kinds of heating methods, you must boil some water for cleanliness. Maybe you can forgo a shower for a bit, but you still need to wash your hands in hot water. You still need hot water to wash dishes and clean up. You especially need hot water when gathering wild edibles or harvesting meat or fish.
This means probably using a rocket stove outside to boil it, since that method produces a small intense fire with minimal wood necessary. If you're advanced in technology and skill, then boiling it inside on a rocket mass heater.
Here's one method that minimizes hot water use, and then the water is collected and used in the garden. That soap won't hurt the plants and may repel bugs. It definitely will organic refuse from the plates, pots and utensils.
A temporary scouring device can be made from the bristly pine tree in your neighborhood. Switch it out often and burn it, because it is loaded with bacteria.
Nothing was wasted on the frontier especially time. Let dishes soak and let the hot water and soap do the trick.
For rinsing, fill up the other side, and place a drop of bleach in it. Your cleanliness has diminished, and you want to ensure no bacterial residue. This will sanitize it.
The old original way of drying dishes was to open up the cabinet above the sink, and then place them on wire racks vertically so they would drip the water back into the sink. Like this:
[link to 4.bp.blogspot.com
It can be freestanding too like this:
[link to 1.bp.blogspot.com
You sure don't want more laundry, so you don't want to mess with dishtowels, do you?
Next, you'll open up the drain and funnel all the water into a bucket. If the toilets (sewage) are working, then you could use that wash/rinse water for flushing. If not, it's into the garden.
This is way faster and uses far less water. Otherwise you can spend and endless amount of time hauling water from your well or some distant river, chopping tons of extra firewood, and waiting for it to boil. It's up to you. Personally, I'd like some time to go trap a squirrel.
Washing dishes in a camping situation, most likely on a hunting/trapping expedition overnight, will be a little different. Bring two bins and a large cooking pot for boiling your water. If not cooking on a well burning fire, you will make a lot of soot from its inefficient burn, and make work for yourself. A lot of smoke too. Dumb.
Using a little wood ash first and coating it wet on the pan, will make it easier to come off. That wood ash will also generate lye when a tiny amount is mixed with the hot water.
Use a little fine grit sand to scrub pots. What this means is that you screwed up and let food burn (and go to waste) on the fire that was too hot.
There's always some dingdong that will try to wash the dishes directly by the water. Your water goes somewhere downstream. This is a stupid idea. Ever see the scene in Kingdom of Heaven. The priest/Hospitaller Knight is trying to brush his teeth. The Moorish Crusader Knight is trying to wash up, and the dingdong squire is urinating just upstream. Someone very dim-witted person is always upstream dumping into the river.
You want to bury your rinse water and disguise your activity, for you'll attract others to your site after you leave if you don't do this. Leave as little a trace as possible.
If you don't put away your rinse water, then of course a raccoon will come callin' since he/she wants to rinse off their food. Many a camper has discovered this.