Remove any appliances from near the kitchen counters. Disconnect the sink and remove it from the current countertop, then disconnect the current countertop from the cabinets and pull it up. Some countertops may attach with screws or nails, and you may need a crowbar to get them up.
Use the old countertop to draw an outline on your plywood. Cut the plywood pieces to the same dimensions as the old countertop, including the cutouts for the sink.
Cut pieces of cement backerboard using the plywood as a template. Score the backerboard with your utility knife, then bend the board to break it along the score mark. Use a jigsaw to cut the sink opening.
Place the plywood over your counters and screw it into place along the front and back edges with the 1 5/8-inch screws.
Mix up your thinset mortar and apply it to the plywood. Use your notched trowel to create ridges in the thinset. Place your backerboard over the thinset and screw it into place with the 1 1/4-inch screws. Place a screw about every 8 inches in several rows running down the backerboard.
Place the tiles on top of the backerboard and use spacers to simulate the grout lines. Make measurements to determine which tiles you will need to cut, then mark those tiles with a pencil.
Place one of the marked tiles into your tile cutter and align the center of the cutter with your pencil mark. Press down on the tile cutter handle to cut the tile. Repeat until you have cut all the tiles that you need to cut.
Use your notched trowel to spread a ribbon of thinset mortar around the outside edges of the top of your counter. Apply tile mastic to the sides of the countertop. Press your edge tiles, those that cover both the top and the sides of the countertop, into place into the mortar and mastic. Use tile spacers to keep the tiles a uniform distance apart.
Spread mortar over about a 6-square-foot area of the countertop. Go over the mortar with your notched trowel to create ridges. Put the tile into place in this area of the countertop. Use spacers to ensure that the tiles are correctly spaced. Once the tile is straight, apply the mortar to another area of the countertop and repeat the process. Continue until all of the tiles are set.
Let the mortar set according to the instructions on your particular brand of mortar. This can take anywhere from a few hours to a full day.
Mix the grout according to the directions that come with the grout. Place the grout onto the countertop and use your grout float to work it into the gaps between the tiles. Run the grout float across the tiles in a line perpendicular to the spaces. When the grout has filled in all the spaces, run the float diagonally across the tiles to remove the excess grout.
Take a damp sponge, not a dripping wet one, and start wiping down the countertop to remove the haze of the grout. Clean the sponge frequently to avoid reapplying the grout. Continue until the countertop is finished.