Always wash your dishes with safe soap like Bio Suds and dispose of your waste water away from shoreline and in the back woods away from campsite. As well I burn waste food in campfire. 

You can use needles from a tree to wash your dishes instead of using a sponge. Also use bio soap and put water in the woods or in the fire pit

I scrape my plates meticulously with a rubber scraper and add the waste to my garbage bag that gets packed out. Then I wash with a few drops of an approved biodegradable soap. The waste water is carried well away from the camp site and away from water, and buried

When I am in the backcountry I have a few steps I do to clean the dishes. Firstly I do my best to eat every crumb, sometimes I even lick my plates, and cutlery.  Then I put any leftover foodbits (stuff I may not have liked or been able to eat) in the garbage ziplock that stays in my bear vault).  Then I wipe the dishes/cutlery that no longer has any food bits on it with a paper towel (which can be burned or put in garbage bag in bear vault) or a j cloth, that i also keep in the bear vault.  Lastly, once there is no food residue visible, I rinse the dishes/cutlery/etc with water.  All of this should be done 50-100 feet from your campsite and all food bits should be carried out with you.  Not sure if anyone mentioned above, but biodegradable soap is not good to use in the lake, it is only biodegradable on land and should always be used a great distance from any water source.  Hope this helps! 🙂  Happy Camping!