I can't remember where I read this, but I was going through a similar
situation and I read that bleach will remove most mold & mildew from the
surface, but LYSOL is one of the few cleaners that will kill it off down
deep. The nice thing about LYSOL is that it is a cleaner also, so it
will get rid of contaminants other than mold which could interfere with
a good caulk job. Bleach has no soaps or surfactants in it to break up
grease & stuff.

I got a tip from a pro when I messed up caulking my tub enclosure. He
told me to wipe the area with full strength bleach, let dry, before
caulking. I did that and all was fine. I made sure there was no soap
scum and I usually give a wipe with denatured alcohol before the bleach.