Update: We moved and our new place has a dishwasher. I don't recycle water much at all anymore, except maybe the rinse water, which I reuse in our tiny garden. I recycled once, maybe twice since we've lived here, when I felt compelled to wash dishes by hand; it is infrequent. I do, however, use the energy saving mode on our dishwasher, and I don't use heated drying, opting instead to open the dishwasher up when it's done and let it air dry. Usually that's not quite enough, especially for plastics, so I dry it with a towel before putting the dishes away.
The key to capturing grey dish water is to capture only the rinse water, and not to rinse the dishes in the already captured water.
It's also a good idea to clean the bucket periodically. Once a week with less than a teaspoon of bleach should do the trick.
Also covering the bucket when not in use will ensure mosquitoes can't breed, even if they do manage to find a way inside your home.