May not have been pure silicone - there are many compounds that use silicone as one component but function as adhesive or caulks. These days there are even silicone-based caulks that are paintable (GE Silicone II, for example). The net is that one must be careful about using the term "silicone" loosely.
In this application, use a caulk that is formulated for kitchen & bath. The old "silicone" should peel or scrape off and a putty knife, scraper blade or coarse cloth or cleaning/polishing pad (even a Scotchguard Dobie!) will aid in the process. Once removed, you can wipe with alcohol or mineral spirits if you think there may be an oil residue from the old silicone. Be careful of using lacquer thinner, acetone, MEK, etc unless you are sure it won't effect the countertop or sink (if the sink is a plastic type).