Turn off the water supply. Disconnect the water pipes from underneath the sink, using a wrench. Remove any nuts holding the faucet in place. Slide the faucet out of the sink. Disconnect and remove the drain. Remove any nuts holding the sink in place using the wrench. Remove any caulk or other adhesive connecting the sink to the counter, using a sharp knife. Remove the sink from the counter.
Clean the sink thoroughly using a cloth soaked in acetone. Scrub away any tough stains and marks with a scouring pad. Sand the sink using the sandpaper to lightly scratch the surface of the metal so it will take the paint. Clean the sink again using acetone and a cloth to remove any debris.
Spray a primer all over the sink, giving the surface as even a coat as you possibly can. Leave the primer to dry for a few hours. The precise amount of time required will depend on your brand of primer, so always check the manufacturer's instructions. When the primer is dry, dust the sink with a dry cloth to remove any loose particles of primer, as these will interfere with painting if left behind.
Apply the oil-based, black enamel paint to the sink, either by spraying it on if it is spray paint or by applying it with a paintbrush. Wait for the paint to become tacky, then apply a second coat. Wait 24 hours for the paint to dry fully. Re-attach the sink to the counter and plumb it back in. Apply a new layer of caulk or other adhesive to fix it to the counter.