How do you re-caulk a tub with mildew behind old caulk?

This is pretty straight forward.

Remove the old caulk. Use a hard rubber scrapper to do this so you don't scrap the tub finish.

Clean the area with a mildew cleaner, any will do, check your local stores. After that, clean the area with mineral spirits to get any grease or soap residue off.

Make sure it's completely dry. Apply a 100% pure silicone to the seam. Smooth with your finger or anyone of the tools out there they sell. Let this dry for 24 hours before you get it wet.