Cut out the old caulk with a utility knife. Trim along the sink where the old caulk was located. Scrape out stubborn caulk with a putty knife or a paint scraper.
Mix bleach and water mixture. The mixture should be one part chlorine bleach with three parts warm water. Use the mixture to clean mold or grime from the area. Allow the mixture to dry completely before applying the painter's tape.
Create a template for the caulk with painter's tape. Place the painter's tape on either side of the area to be caulked. The space between should be approximately 1/4 inch. Apply a bead of silicone caulk in the 1/4-inch space in the painter's tape template. Use a finger to evenly spread the caulk smooth. Remove the painter's tape immediately for a clean line.