I think the best time to go is spring or autumn. Summer is way too hot and it can get pretty cold at night in the winter.
If you're going for the first time, personally I'd recommend going with a good specialist bellydance tour - otherwise you can miss out on so much and also Cairo is so full on you might find it a bit much.
The first time I ever went (in 1980) I went on my own and it was pretty intimidating. I saw a lot of dancing - it was a real peak time for cabaret clubs - but didn't know where to go to find good costumes and I know I missed out on an awful lot of other things.
the second time I went (just a few years ago) I went with Farida Adventures
www.faridaadventures.com and it was just fantastic - we did everything a bellydancer could ever want to do.
We saw loads of great shows, had a front row table at Dina's show, went shopping at all the top designers as well as the Khan el Khalili, saw the tannoura, had a party night with live musicians and great folkloric dancers at Yasmina of Cairo's apartment and and and....it was just brilliant!
Since then I've gone back on my own and twice again with Farida - Farida gave me the confidence and knowledge to go on my own but I also get something new every time I go with Farida. I can't recommend them highly enough