Hang the tiles for the backsplash and grout the lines between them. Don't grout the outer border of the tiles, where they meet the wall, and don't grout the bottom space between the tiles and the countertop. Let the grout set for 48 hours.
Mix a solution of one part bleach, five parts water in a spray bottle. Use it to soak the perimeter of the backsplash, including the open line between the backsplash and the countertop. Scrub it with a stiff toothbrush. Rinse well. Let it dry overnight.
Load your caulk gun with its caulk. Cut off the end of the plastic tip of the caulk container with your razor knife, making the cut at a 45-degree angle and making the hole about ¼-inch across. Jam a nail into the hole to break the interior seal. Squeeze the trigger of the caulk gun until caulk comes out of the end. Trip the release latch at the back end of the gun to stop the flow of caulk.
Place the tip to one end of the line where the tile meets the countertop. Squeeze the trigger, expelling caulk into the line. Pull the gun slowly across the length of the lines as you continue squeezing the trigger, so it leaves a smooth bead of caulk into the whole space. Stop the flow of caulk at the end by tripping the release latch at the back of the gun.
Wet your thumb under the water faucet. Run your thumb along the surface of the caulk line, from end to end, pressing gently to set it into the line.
Repeat the process for each edge of the backsplash. Let the caulk set for 24 hours before using the sink.