how to clean caulk between tub and tile
Removing caulking can be time consuming, especially in the bathroom. From my experience there is no magic solution.

I recommend these tools:


flat head screw driver
small paint scrapper (plastic as to not scratch any finishes)
hair dryer


Using the hair dryer, heat up the caulking, then use paint scrapper to loosen the caulking by pushing on top and bellow, try to keep the caulk intact. Once you've freed one end, try to gently pull away following the line in which is was laid. Use the paint scrapper as you peel away and try to get long lengths to keep it clean. 

The flat head screwdriver is if you have difficulty getting a started but usually the paint scraper will do the job.

If the caulking is dry, you will be using both the screw driver and paint scraper.

Patience is a must.