Position the dishwasher exactly as you want it once it's enclosed between the kitchen cabinets and the false wall. Allow a slight gap between the dishwasher and the cabinet to ensure proper clearance later.
Hold a carpenter's square flat against the wall just above the dishwasher's top. Allow one end of the square to hang past the dishwasher one-half inch to 1 inch. Run a line down the end of the square to notate the clearance necessary.
Pull the dishwasher back out again. Make sure it's far enough away that you can work in the area easily.
Measure over from the cabinet to the clearance mark on the wall to determine how far away the false wall will rest from the cabinet structure. Use this measurement to mark the wall again near the bottom of the wall, as well as in two or three places on the floor. Use tape or a chalk line to make the floor marking more visible.
Mark the floor to indicate the end of the false wall. Record this measurement as the wall length. Typically this will match either the depth of the cabinetry on the other side of the dishwasher or the depth of the dishwasher itself.
Hold a level against the wall, aligned with the marks. Double-check that the level reads plumb -- straight up and down -- before marking a straight line through the guide marks.
Cut two boards -- either 2-by-4s or 1-by-2s -- to a length equal to the depth you intend the false wall to be (the distance from the existing wall to the end of the new false wall). These will form your top and bottom wall plates, boards that will horizontally sandwich the false wall's vertical studs. For a wall corresponding to the depth of the cabinets, for example, you might cut the plates 24 inches long -- the depth of a typical cupboard.
Measure from the floor up to the height desired. Countertops are typically about 36 inches high, but you can build the wall slightly taller or shorter.
Subtract the thickness of the plates used -- 1 1/2 inches per plate for 2-by-4s but 3/4 inch each for 1-by-2s -- to determine the necessary stud measurements. A 24-inch-high wall built from 1-by-2 boards, for instance, uses 22-1/2-inch-long studs.
Cut one stud for each 8 inches of wall length, plus two extra studs to allow for double studs at either end of the plates. The extra framework will help make the wall sturdier.
Line the plates up side by side, perfectly even. Measure over from one edge to the 8-inch mark. Lay a square across the boards and draw a line through both at once. Repeat as necessary until you reach the opposite end of the plates.
Lay a plate on the thin edge so the wide face of the board faces you. Align a stud with the end of the plate. Nail through the plate into the stud with two nails. Skip to the first 8-inch mark and repeat with another stud. Work across to the end, nailing a final stud flush with the end of the plate.
Flip the frame over so the free end of the studs face you. Line the opposite plate up and nail through the plate into each stud again. Ignore the remaining two studs, which install next to the first and last stud, for now.
Pick the stud wall up and set it in place according to the wall and floor guide marks you previously made. Hold a level up against the end of the wall to check plumb again. Adjust as necessary.
Drive the first nail through the bottom plate into the floor as near to the wall corner as possible. Repeat with an additional nail to the side and slightly below the first nail to create a staggered attachment pattern. This provides additional wall stability.
Continue across the wall, nailing every 4 to 6 inches. Pound the last two nails as close to the final stud as possible.
Nail through the first stud into the wall every 6 inches. Stagger the nails as you did previously.
Insert the remaining two studs, pushing them against the first and last studs. You may have to tap them slightly with a hammer; sanding the studs just enough to slide in place is another option. Toenail the boards -- driving the nails at an angle -- through the boards into the plates as well as straight through the studs into the adjacent stud.
Insulate the wall, spread a vapor barrier cut to fit on the side facing the dishwasher, and finish as desired. The wall requires covering on both sides as well as the end of the wall, formed by the final stud, and the top. Push the dishwasher back in place when complete.