Measure the depth of the cabinet that will be next to the dishwasher. Then, measure the width of the dishwasher. Usually, cabinets are 24 inches deep and dishwashers are 24 inches wide, therefore, the outside of the frame should measure 24 inches by 24 inches.
Create a frame to support the countertop using 2-by-2s, wood glue and wood screws. Allow the glue to cure.
Locate the wall studs in the wall behind which the dishwasher will be installed. Attach the frame to the wall studs using screws; then, attach it to the adjacent cabinet using screws at regular intervals. Be sure the top of the frame is level with the top of the adjacent cabinets.
Cut the MDF to create an end piece that will hide the side of the dishwasher and support the outside edge of the countertop. The depth and height of the piece match the depth of the lower cabinetry, in this case, 24 inches deep by 34 1/2 inches tall. To better support the frame and counter, the MDF will be under the 2-by-2. This means the MDF should be cut to allow for the depth of the 2-by-2, which is 1 1/2 inches. The final size the MDF in this case will be 33 inches tall by 24 inches deep.
Put the MDF in place under the outside edge of the frame and attach with screws. Use a level to ensure the MDF is straight, then attach the bottom of it to the floor, driving screws into the MDF at an angle to penetrate the floor. Finish the MDF by painting it to match the kitchen cabinetry.
Attach trim to the outside edge of the MDF using brads or finishing nails. The goal of this is to finish the look as well as hide the screws.
Calculate the width of the new piece of countertop. Typically, countertops overhang the sides of lower cabinets between 1 and 1 1/2 inches. To determine the width of countertop, add the width of the frame to the depth of the MDF to get the measurement of 24 3/4 inches. In order to have the countertop overhang the MDF by 1 1/2 inches, add 1 1/2 to 24 3/4. The width of the countertop should be 26 1/4 inches.
Lay painter's tape along the counter where it will be cut in order to prevent chipping. Measure and mark 26 1/4 inches at regular intervals. Then, use a straightedge and a pencil to draw the cut line on top of the tape.
Cut the piece of laminate countertop to size using a circular saw. Remove the tape and wipe the dust off the surface.
Place the countertop on the frame. Attach it with wood screws from the underside of the frame. Be sure the screws are long enough to penetrate the underside of the countertop but not so long they will come through the top of it.
Apply caulk to the seam between the new piece of countertop and the pre-existing countertop; smooth the caulk line.