Scrape off old grout with a chisel. All of the old grout will need to be removed. Get under the lip of the sink with your chisel if your sink is an over-mount model.
Tape off the sink and the counter with painter's tape, leaving a quarter inch space between the strips. This will make your grout edges clean.
Place the tube of grout inside the grout gun. Cut the top off the tube and ensure it is inserted firmly.
Lay down a quarter inch bead of grout around the sink. Work slowly. Do not over-grout.
Smooth out the grout with the back of a spoon or a finger. The surface of the new grout should be even with the tape.
Remove the tape immediately. Do not wait for the grout to cure or the tape will be permanently adhered.