Get a j cloth or similar and dampen it very slightly. Dip it into some bicarbonate of soda and use it as a scourer.
I wouldn't use bleach as I used to work on a ward where one of the staff bleached the cups. She "forgot" to rinse mine properly - by accident or otherwise I never knew (she disliked me intensely) but I can still taste it now.
Bleach for toilets only imo.