Remove the old fabric from the chair. Pull out any old staples with pliers, and use the scissors and flathead screwdriver as needed to remove difficult parts of the upholstery. Try to keep the upholstery intact, as you can use it as a pattern for your new upholstery.
Label each piece of fabric as you remove it with a permanent marker. Draw an arrow on the fabric to indicate what part of the fabric goes up. Cut off the seams of each piece of fabric.
Lay the fabric pieces over the new fabric spread over the floor or on a flat table. Draw the outline of each piece onto the new fabric with chalk. Write the name of the piece on the new fabric as well using sticky notes. Add ½ inch to all seam edges for a new seam allowance. Cut out the upholstery pieces.
Sew the back pieces of the chair together on the sewing machine. Go slowly, and follow the directions you outlined on the sticky notes. Sew the front and back of the chair together, along with the arm pieces.
Fit the back and arm pieces over the back of the chair. At this point the upholstery should look like a slipcover. Check the fit and make and necessary adjustments with the sewing machine.
Pull the front back of the fabric down toward the seat and staple the fabric in place. Pull the arm pieces down toward the seat and staple as well. Turn the chair upside down and staple the bottom edges of fabric to the bottom of the chair.
Staple the front piece of fabric that covers the front bottom of the chair in place along the seat and bottom.
Cut a seat cover to cover the rough edges of the chair upholstery and the bottom edge under the chair seat. Tuck the edges of the covers under. Staple the seat cover over the bottom seat of the chair and under the chair seat.
Use the original cushion as a pattern and cut the fabric for a new cushion. Sew the cushion on the sewing machine and turn it right side out.
Stuff the cushion into the new cushion cover and hand sew the opening closed. Replace the cushion inside the bucket chair.