i can't tell if you are landlord or tenant
if tenant - call your landlord immediately, you shouldn't have even attempted without notifying them. don't expect them to be very happy either.
if landlord - you need to find out how the water started the problem and fix that too.
bleach works well for black mold, but you have to be very thorough. i've used it before. i just put cleaner like mr. clean in a bucket of water and add a cup or more of bleach. then i use a rag and just start washing and drying, washing and drying, washing and drying until no sign of the mold is left.
we have a wet basement that this is a problem in if we don't keep it properly dehumidified. i have concrete, which is much easier to deal with. you probably have wallboard. there is a fine line between cleaning, and getting the wallboard so wet you ruin it.
best best would be to completely replace all the wallboard if that is an option. you will have to consider the wall timbers too. they can also be washed with the bleach, but will need to be replaced if not washed.
when i finish cleaning every corner, nook and cranny thoroughly, i spray it down with lysol for mold.
good luck, i don't envy the task you have ahead.