You can also use vinegar or caulk remover to remove old caulk by applying it and letting it sit for several hours for another headache-free removal technique.
Once the caulk is loosened, it should come off easily. Clean the entire space with a bleach solution or ethyl alcohol, then wait for it to completely dry before applying caulk.
Use a caulk gun and cut a 45-degree angle at the tube's tip. Drag the caulk along the gap, use a wet finger to smooth out the line, then wipe off any excess with a wet paper towel.