Turn your stool over to access the staples that hold the fabric in place. Pry up and remove any staples, tacks or nails, using the screwdriver. Remove the old seat cover.
Lay the old seat cover on the upholstery fabric and trace around it, using the tailor's chalk. If the old stool cover is comprised of more than one part, take it apart, using the seam ripper. Trace the individual pieces and cut out the shapes.
Pin the pieces together, following the pattern of the original stool cover. Most stool covers are comprised of a long, skinny piece that forms the "side" of the stool, and a round "top" piece. Fold the strip piece in half width-wise with the outside of the fabric facing inward. Pin the edges. Sew the short edges together, using the same seam allowance as the stool. Remove the pins as you sew. Leave the loop with the wrong side facing outward. Pin the top to the loop, around the edge. Sew around the edge, removing the pins as you go. Turn the piece right-side out.
Remove and discard any seat padding on the stool. Scrape it off with the screwdriver, if necessary.
Lay the new seat padding on your work surface and turn the stool upside down on it. Trace around the top of the stool, using tailor's chalk. Remove the stool and cut your padding along the chalk line.
Spray the top of the stool with spray adhesive. Place the padding on it. Allow it to dry. Pull the fabric seat cover over the top of the stool. Tuck the edges under the bottom of the stool. Staple the fabric to the bottom of the seat of the stool. Cut off any excess fabric.