I hope this doesn't sound gross but I remove all the litter, fill the tub with very hot water, dawn dish soap and some bleach, and let the box soak while I go wash down the litter area including baseboards and walls. I use a small brush and scrub it including the corners and under the rim, then drain the tub and rinse it well with really hot water. I use an old towel saved especially for the purpose to dry it. I put in new litter and while the kitty is christening the new, clean box (why do they do that?), I scrub my tub with cleaner and bleach, hot water and wipe it down.
I use flushable litter so no spare bits of litter get in the drain, and since I scoop three times a day the box is not very messy and really easy to clean. I am not thrilled about using the bathtub but it is the only large space for cleaning I have in the apartment - I don't have a laundry tub or garden hose or anything. And I know how clean I get my tub since I started doing this - no half-assed cleaning job anymore!