maybe an old question, but one I have never found a really good answer to!
I use these backgrounds & they inevitably do get grubby. Its mainly polish from shoes, but there will be alsorts or crap on it. I used to replace mine, quite often when times were not so hard, but nowadays have to resort to cleaning. washing up liquid in water has no effect whatsoever. I have used a washing up scourer with cellulose thinners. but the thinners stinks the studio out for a week! Also a bit hard on the knees! Any bright ideas?
I use Big Wipes Industrial(Makro usually sell them in a big yellow/black tub) on all the vinyl's(on the non-white vinyl's it can leave a residue but wipes off straight away when still damp) http://www.bigwipes.com/
Get your hands on some Isopropanol or Isopropyl. (one is a dilute of the other)
It will clean most plastics and without marking, and dries without residue, and it also anti static. You can also wipes impregnated with Isopropyl.
RS sells this gear, but is cheaper elsewhere. (Halfords sell it also as brake cleaner/degreaser)
For the less ingrained marks I find a standard pencil rubber works well. I've also found the recent rain has been usefull for cleaning off the general grime from a doing 'booth' at a restored railway !. I just left it on the lawn for a morning and let the rain do its thing.