I know - you are PRETTY sure you know how to do dishes. And you are probably right! But like almost all things camping vs all things home, there's a trick to it.
Pictured: Step #1 - water and soap
As soon as I finish the cooking, I take paper towels and wipe leftover food out of the pots and pans, etc. You have no drain, so you want as much of the residue off the dishes before you begin.
I keep my bottled water for drinking and cooking, and use camp faucet water for dishes. I half-fill my dishpan with water from the camp faucet, and I fill my coffeepot as well. I start the pot of water boiling after I'm done with both my meal, and my hanging out by the fire (which I do after both breakfast and dinner - I am big on hanging out by a campfire.)
I use a fresh sponge for every camp trip, but cut in half - you don't need a whole one, and that stretches your sponge budget :)
While the water is heating to a boil, I squirt some dish soap into the water, wet my sponge and get soap onto my sponge as well. Soap choice is important... you are out in the environment, and you want to be sure what you use is biodegradable. I like Mrs. Meyer's Clean Day, and I buy up the spring grass scent, which goes nicely with camping (clearly just a preference... any scent is fine!)
Pictured: Mrs. Meyer's is my favorite dish soap
While the water is working on boiling, get your dishes out on the bench or wherever you are washing, proximate to your dishpan. Also lay out dish towels or paper towels that you'll be placing your washed and rinsed dishes on to dry. This is usually a bit of a juggle for me, using the picnic table or if a large log is nearby, that also.
When the water boils, pour half of it into the dish pan with the cold water and soap, and swish with a utensil to mix. You should have nice warm/hot water for washing.
Pictured: As you wash, place your sudsy dishes on the picnic bench or section of the tabletop.
I tend to put in eating and cooking utensils last, and let them soak a little. The water will get a little dingy... you are out in the wilderness, this is okay. So long as your STUFF is clean, the water may not be.
Once the dishes are washed, ideally, take your water to the rock-lined drainage under the camp faucet and dump, rinse the dish pan, and again, fill half full with cold rinse water. Add in the last half of the hot-now-warm water from the coffee pot, and swirl.
Dip your washed, sudsy waiting dishes into the water to rinse, and set onto the towels or paper towels you laid out earlier for this purpose. You can balance the dishes offset slightly on the towel to let air circulate and air dry. You COULD towel dry them... but I just usually want to do other things, like hammock or wander a bit. So I let the forest air dry them, dump my rinse water and upend the dish pan, and off I go!