After straining with burlap (14 threads per inch is what I just counted with my magnifying glass) I can't imagine straining with as fine as a paint filter or pantyhose.
Kitchen dishwater would clog those up in no time. One MUST HAVE a garden sprinkle can, about $2-$4, to spread the greywater on the road after a little chlorine added and let sit.
The system we used in a kitchen feeding 20 was very efficient:
----get two 5 gallon paint buckets
----on one of them, cut it in two, the useful part is the top part, with the handle
----cut out MANY pieces of burlap, about 2feet x 2feet square
----lay burlap on top of "whole" 5 gallon paint bucket
----force "cut" paint bucket down into "whole" paint bucket, which will tighten burlap like a drumhead
----fill with a gallon or two of nasty water and let sit. If it jams up, use something to scrape the burlap
----don't bother to wash the burlap, just dry it in the sun and toss it. Burlap is cheap and you won't need more than one a day if you are a small camp. Less if you are just a few people.
-----treat "whole" paint bucket filtered greywater and spread on road with cheap garden sprinkle can
The burlap does a great job. The particles that come thru are very small and won't ever clog the sprinkle can.
If you are really worried about the small particles, just GENTLY pour most water that has been silently sitting and letting the particles settle to the bottom, and when those particles start showing up as you're pouring into the sprinkle can, just get a FUNNEL and pour it into one of your empty water bottles and take it home.