The best way to remove caulk from around your bathtub is to use an inexpensive caulk scraper made by a company named Komax. This tool is so great because it actually works (due to its pointed V-shaped edge which helps lift and scrape caulk without damaging your bathtub.)
I actually stumbled across this tool while crusing the aisles at my local Walmart. Finding this tool was such a relief since all of the articles/guides on the 'net' on how to remove caulk from your bathtub were not giving me any results. At a low price of 3 bucks, you will not be disappointed.