Scraping out moldy caulk from around a bathtub is a tough, tedious job. And if you get too aggressive, you'll chip a tile or mar the tub finish. Make the job a lot easier by first softening the caulk with a caulk remover (inexpensive at hardware stores).
Application is simple. Just squeeze the remover onto the caulk, making sure it covers the edges (Photo 1). Let it sit for several hours (or as directed on the package). Then start scraping out the old caulk (Photo 2). The caulk remover works by destroying the bond between the caulk and the tub or tile, so that instead of chipping at the caulk with a razor blade, you just pull away big chunks of it with a putty knife. Most removers work on both silicone and acrylic latex caulks.
Clean off the residue with soap and water, let dry and then recaulk.