You don't want to. Run the car a few laps, and clean the tires (sticky tape, spin them on a sponge or rag, rub with wet fingers, whatever), and keep running. After a few dozen laps and a few cleanings, the track will be as clean as it's going to get. If the tires just won't stay gripping, try softening them up with a quick rub with WD-40. Or, when you're done, rub in some 3-in-1 oil so that next time you drive, the tires are nice and soft.
There's nothing you can do to keep the tires from picking up dust, but a good tire will maintain grip even with a few specks here and there. But if you haven't run the track in weeks, you're better off using a dust cloth on it first anyway. ;-)