There is no way to remove the mildew without first removing the caulk. You can do this with the blade of a razor knife. Once the caulk has been removed any grout that remains below it should also be removed. The sharp end of an old-fashioned can opener works well.

Next, clean up the mildew using a solution of 1/3 cup of laundry detergent, 1 qt. liquid bleach and 3 quarts of warm water. Add the bleach to the water first and then add the detergent. Even though the solution is mild you should wear rubber gloves, eye protection, and have plenty of ventilation. Rinse the area thoroughly with fresh water and dry it out completely. A hair dryer will help dry out the joints. Finally, grout the area. After the grout has been allowed to cure for a couple of weeks, seal the tile and grout with an acrylic or silicone tile and grout sealer.