You can easily kill black mold in the shower by using some distilled white vinegar and a spray bottle.
Take the distilled white vinegar and add it to a spray bottle and spray the vinegar directly onto the mold in the shower and allow it to sit for awhile.
You should leave the vinegar on the mold for around 10 to 30 minutes to allow it enough time to work on killing the mold.
After spraying the vinegar onto the mold use a sponge or a good scrub brush to scrub the mold off the shower and then rinse it off with water.
If the mold is worse and doesn't come off very easy with the vinegar then you might need to use some bleach instead of the vinegar.
Mix 1 part bleach to 10 parts water and put it in a bucket or spray bottle and then apply the bleach to the mold and let it sit for a few minutes and then use the scrub brush or sponge again to scrub away the mold.
You can also buy already made mold killing products for showers but usually the mold in the shower can easily be killed off by using bleach or vinegar.