Removing bathtub caulk can be a time-consuming, messy job. Allow yourself at least a half of a day for the project and wear clothes that you feel comfortable cleaning in.
Clean the bathtub thoroughly to get rid of existing soap scum and mold and let the bathtub dry completely. Then, find an edge of the caulk that is already peeling and wedge a putty knife underneath it to remove the caulk. Take special care not to scratch the tub fixture when handling more stubborn pieces of caulk.
Use a single-edge razor blade tool, like those used for painting, to slice caulk out of smaller spaces. Remove stubborn pieces of caulk from deep crevices with a pair of pliers and clean the remaining caulk away using a bristled scrub brush or hard toothbrush.