Turn the chair upside down and locate the screws and bolts that are holding the chair together. Remove the screws and bolts but remember where they all go. Write it down and label if needed. Set the legs to the side for later.
Turn the seat upside down. Remove the staples and tacks holding the upholstery to the chair. This will take a bit of work. If you come across a particularly stubborn one, see if you can pull the old upholstery off without removing it. It will be alright to leave a few, just don't get carried away. Take note of areas that are folded and pleated, so that you can duplicate it. Do not cut the old upholstery or destroy it, it will become your pattern for the new fabric.
Remove the fabric from the chair back in the same way. Pay close attention to areas that are stitched. You will want to copy that with the new fabric. Carefully take the seams apart with the seam ripper.
Iron the old fabric to get flat pattern pieces. Lay the new upholstery fabric right side down on your work surface. Place the old upholstery pattern pieces on top of the new fabric with the right side down. Pin the pattern pieces to the new upholstery fabric. Cut out the pieces.
Lay the quilt batting on your work surface. Place the old upholstery pieces on top of the batting. Pin and cut out the batting. Position the batting on the chair. Staple and sew the batting in place. This does not need to be perfect, you will not see it. If your chair does not need extra cushion, you can omit this step.
Place the new seat upholstery fabric right side down on you work surface. Place the seat upside down on the fabric. Pull up the middle of one side of the fabric and wrap it to the bottom of the chair. Staple it once. Turn the seat to the opposite side. Pull up the middle of the fabric and wrap it to the bottom of the chair. Staple it once. Repeat for the other two sides. Continue to staple the rest of the fabric to the bottom of the chair. Pleat and fold where needed.
Replace the upholstery fabric on the back of the chair working backward from the way you took the old upholstery off. Fold under the edges on the side of the chair back and hand sew with an upholstery needle and sturdy upholstery thread. Sink the stitches into the folds of the fabric to hide them.
Sand and stain the legs. Allow the stain to dry. Reassemble the reupholstered fabric chair.