You need to:
1. dissolve the clogged surface of the grout with an acidic cleaner.
2. Then, if there's mildew growing in the grout, then apply bleach straight out of the jug.
Typically, people have trouble cleaning mildew off of grout because the bleach doesn't penetrate into the grout, and the reason for that is that the surface of the grout is all clogged up with dirt and soap scum.
Probably the best acidic cleaner for removing the soap scum is a phosphoric acid based toilet bowl cleaner (about 15 to 20 percent phosphoric acid). The reason why is that it's gelled to stick to the bowl better, and that also means that it'll stick to vertical surfaces better. Also, cuz it's gelled, it won't get wicked into the grout the way CLR would; it'll just sit on the surface of the grout.
So, the first step is to clean the surface of the grout and remove all the soap scum or dirt clogging up the surface.
Then, apply the bleach to remove mildew, if any.
Then, if there is mildew, but the bleach isn't removing it, you probably need to hit it again with the acid cuz the surface is still clogged.
Also, allow a full day to dry after cleaning before applying bleach. (at least a full day, that is)