How do I clean caulk around the kitchen sink?

The caulk should easily clean up with any general cleaner. IF you have black stains it's likely that there is water behind the caulk. In this case it should be removed and new caulk put on after a thorough cleaning and allowing it to dry completely. Check to be sure there is no water leak anywhere.

Make a paste of Comet and water, scrub with an old toothbrush, rinse well.

Here is a link on how to replace. I just use Dawn original and it gets cleaned a bunch of times throughout the day. I also dry everything off with a dish towel or paper towel in between uses.

http://m.wikihow.com/Caulk-the-Kitchen-Sink

Are you talking about the grayish-brownish plumber’s putty under the rim of a drop-in sink? Kind of soft and clay-like?

Youre kind of stuck with it. It looks like a many year collection of dirt and grease. But it keeps water from getting into the cabinet below.

Or the caulk that is usually white although it comes in other colors, including clear and is firmer and kind of plasticky?

If so, there are some good suggestions above.

If it's not too bad, use Barkeeper's Friend. Sprinkle, scrub, rinse, voila! If it's bad and looks moldy, remove it, clean the area, then recaulk with new kitchen caulking. Best of luck.

http://homeguides.sfgate.com/replace-seal-around-kitchen-sink-51278.html

Scrape it off with a single razor blade tool, they have them at the Dollar Tree.... https://www.amazon.com/dp/B01HNGSWAS/ref=asc_df_B01HNGSWAS5133010/?tag=hyprod-20&creative=395033&creativeASIN=B01HNGSWAS&linkCode=df0&hvadid=198077680239&hvpos=1o4&hvnetw=g&hvrand=14235305049061091291&hvpone=&hvptwo=&hvqmt=&hvdev=c&hvdvcmdl=&hvlocint=&hvlocphy=9033057&hvtargid=pla-373627482887