You must be patient and thorough when removing silicone caulk to be sure all traces of the old caulk are gone, or you will not be able to get a solid seal with the new caulk.
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Clean the area around the caulk before you start cutting into it.
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Use a razor blade or box cutter to cut along both edges of the silicone caulk, breaking the seal with the wall and porcelain fixture. Make a vertical cut somewhere along the caulk to give yourself an end to pry up from the porcelain.
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Pry up one end of the silicone caulk with a putty knife. Grab the loose end with your hand and start pulling the caulk off of the porcelain. As you pull, use your putty knife to loosen the caulk in front of you. Go slow to make sure you get as much of the silicone removed as possible. It is much easier to pull the caulk off than to scrape it off with a blade.
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Scrape off the silicone caulk that remains with a razor blade or box cutter. Be careful not to scratch the porcelain surface with your blade. Once you have scraped off as much caulk as possible, use a rag moistened with rubbing alcohol to remove the leftover residue.