Washin' dishes
There are many ways to do this in the backcountry. If you have a site with a pit toilet you can wash everything under your water container and catch the dirty water in a pan, then dump all that in the toilet. If you don't have that option you will have to walk a good distance from camp, dig as deep a hole as possible and dump it in there. You really want to keep all the food smells out of your camp. You can also use bins to fill with water and wash and rinse, but that dirties up a lot of water you have to dispose of.

Now, if you don't mind using some paper towels, here is a method that works great and does not use a lot of water.
1. Wipe all the food you can out of dishes with a paper towel.
2. Get your sponge wet(but not soaking wet) and use a little biodegradable soap and clean off anything that is left.
3. Wipe out soapy water with a paper towel.
4. Rinse with enough water to get it wet, then wipe out excess with a paper towel.
5. Completely dry with a dish towel.

helpful hint: if you have an empty gatorade bottle, they are great for stuffing the used paper towels in, or if you have leftover food you can seal it in the bottle till you can throw it away. It helps keep a clean camp.
beyondthecampground.com
Here is our basic "tailgate kitchen" set up. Everything you need for cooking is within easy reach. All the food is in the bin (Rubbermaid actionpacker) next to the ice chest.
backcountry camping for the whole family
step 1
step 2
step 4
step 3
trash
water for dishwashing/ handwashing
ice chest
food bin