Locate a stud in the wall from which the breakfast bar will extend, and mark the location of the stud.
Remove the baseboard trim from the wall with a reciprocating saw.
Cut a sill plate for the breakfast bar wall from a pressure-treated 2-by-4-inch board. The sill plate connects the wall frame to the floor.
Place the sill plate onto the floor, butting up one end against the wall where you removed the trim. Drive galvanized screws through the sill plate and into the floor, using one screw every 6 inches.
Measure and cut two standard 2-by-4 boards that are the same length as the sill plate. These boards will be the top and bottom plates of the framing.
Cut the wall studs for the framing out of standard 2-by-4 boards. Cut the studs to the desired height of the breakfast bar. Determine how many studs you need by dividing the length of the sill plate by 16, as the studs must be placed no farther apart than 16 inches.
Assemble the framing by nailing a stud into each end of the bottom plate, and nailing the other studs in between them. Nail the top plate to the top of the studs.
Place the frame on top of the sill plate, and nail it to the stud in the perpendicular wall and by driving nails through the bottom plate into the sill plate.
Measure the distance from the top of the wall frame to the floor, and subtract 1/4 inch. Cut 1/2-inch drywall boards for the wall to fit that measurement.
Screw the drywall to the wall frame with drywall screws, positioning it so that the 1/4-inch gap is along the bottom of the wall. Drive the screw heads deep enough that they are just beneath the surface of the drywall, which will hide them when the walls are finished. Use one drywall screw every 8 inches.
Cover the outer edges of the wall where it extends into the center of the room by stapling metal corner beads. Use one staple every 3 inches along the wall.
Finish the seams between the drywall boards by covering them with joint compound and pressing drywall tape into the compound with a putty knife. Place the joint compound and drywall tape over the metal corner beads as well. Allow the joint compound to dry overnight.
Sand the joint compound smooth, and wipe the dust off the walls with a clean, dry rag. Apply two more layers of joint compound and drywall tape, allowing each to dry overnight.
Apply a coat of primer to the wall and allow it to dry.
Apply two coats of paint to the wall, allowing each coat to dry.
Nail baseboard trim to the bottom of the wall where it meets the floor.
Nail supports for the countertop -- called corbels -- to the top edge of the wall. Place the corbels over the wall studs to provide added strength.
Install the countertop for the breakfast bar. There are a variety of materials from which to build the countertop, including wood, stone and laminate, and the precise installation method depends on which material you use for the countertop.