For greywater holding in campgrounds, I keep a dedicated collapsible 5 gallon container. I've modified the spout to allow a drain hose to be attached, as some campgrounds don't allow the use of an open container for this purpose.
The biggest problem with greywater in other settings is that if improperly controlled you'll create bug/ smell problems for yourself and the next camper. Not to mention general nastiness if it's just dispersed here and there!
Unless you're big users of strong chemicals, your greywater poses no risk to the environment. Use mild soaps to wash yourself and your dishes- its strength will be expended on the task, leaving you with-dirty water!
Greywater can be properly disposed of in dispersed
camping situations by digging a hole at least 100 ft. from water, placing your drain hose into it, and temporarily backfilling with a few big rocks. Ashes (COLD) from the inevitable fire ring can be liberally sprinkled in for bug control. The use of a simple filter as suggested above is a good idea.
The hole must be filled and tamped before leaving the site.
The same protocol governs latrine holes, if you need them, although they should be deeper, especially if you put T.P. in it- animals like to dig the paper up. Cold ashes are also an excellent bug/odor control for a latrine hole.
If you use NO chemicals and put NO toilet paper in your porta-potty, you can dispose of its contents in a deep
latrine hole, as before well filled and tamped upon breaking camp.
If you're a minimalist that uses small amounts of water for dishwashing- or a tent camper- the best place to pour the dregs is into the fire ring. They're usually many inches thick with ash, which is a very efficient natural filter. If you're using the fire ring at the time, just pour it in at the edge- it won't extinguish your fire. Any food bits will be consumed, and won't attract bugs and/or rodents.
Make sure when you break camp to thoroughly mix the fire ring ashes so that the next camper will have a clean place to start with.
And have fun!