Unscrew and remove the existing countertop from the cabinets with a screwdriver. Work carefully not to damage the countertop so that you can use the countertop as a template for the new tile countertop.
Measure, mark and cut the three-quarter-inch plywood base for the cabinet top. Use the measurements from the existing countertop to do this. Cut the plywood with a circular saw. Screw the plywood base to the cabinet blocks just inside the cabinets. If your cabinets do not have blocks, you will need to cut small blocks from a 2-by-4 board and glue them to the inside of your cabinets with wood glue.
Transfer the same measurements you used on the plywood to a cement backer board with a pencil. Score one side of the cement backer board with a utility knife and snap the backer board from the other side. Screw the cement backer board into the plywood with exterior grade screws and a screwdriver. Cover any gaps between cement backer board pieces with fiberglass mesh tape. Spread thin set mortar over the mesh tape with a trowel. Give the mortar 24 hours to dry.
Find a layout for the tile that you like. Arrange the tile on the top of the cement backer board base until you are happy with the layout. Make sure you place tile spacers between the tiles to account for grout lines. Mark the starting spot for the tile layout on the cement backer board with a pencil.
Install the tile on the cement backer board. Spread thin set mortar with a notched trowel. Begin at the marked starting point and spread small sections of mortar at a time. Press the tiles into the mortar and place tile spacers between the tiles. Use a tile saw to cut the tiles to size. Give the mortar 24 hours to set up prior to grouting.
Pull the tile spacers from between the tiles. Remove any excess dried mortar between the tiles with a utility knife. Vacuum the countertop to remove debris and dried pieces of mortar.
Scoop a small amount of grout onto a rubber grout float. Spread the grout over the countertop tiles in a diagonal motion. Fill the tile joints and then remove the excess grout from the tiles with the float. Wait 20 minutes for the grout to set up and then use a damp grout sponge to wipe away grout from the tile tops. Allow the grout to dry overnight. Use a dry rag to buff the grout residue from the tile.