As a bathtub re-finisher, I am constantly asked how to remove mold and mildew from around the tub and within the grout lines on the tile surround or inside a shower. It is actually black mildew that we are fighting and it’s not an easy fight.
Mildew binds itself to porous areas and areas that give it a surface to cling to such as silicone caulk. When you have mildew on the caulk between a bathtub and the surround it is best to remove it and just start over with a non-silicone caulk with a mildew resistant micro-ban additive.
Take a utility knife and score the old caulk length-wise from the top and from the side repeatedly until you have the majority removed. Then take your window scraper razor blade and go after the rest. Before you caulk it back in get all the nearby mildew (on the tile grout) removed as well.
Mildew thrives in a higher PH environment. Soap and minerals have higher PH levels and thus encourage mildew to grow. Staying on top of soap and minerals is the way to keep mildew at bay and this is done with a more acidic, lower PH cleaner such as Barkeepers Friend, which I highly recommend.
The key to removing mildew is by keeping a strong bleach solution on top of the affected area for a good amount of time. Having a strong grout brush is also important to impregnate the solution into the grout and work the mildew loose.