Flip the chair over and remove the screws that hold the seat in place. Set the screws aside in a safe place.
Turn the chair back over and remove the padded chair seat.
Remove the staples that hold the fabric to the chair seat. Use a staple remover or a flat head screwdriver. This will leave the padded chair seat as well as the padding and some gauze.
Lay the replacement upholstery fabric wrong side up on your workspace. Cut the fabric with scissors so that it is at least 5 inches larger on all sides than the chair seat.
Iron the fabric to remove any creases.
Lay the fabric face down on your work surface. Place the seat, with the padding still in place, face down, centered on top of the fabric.
Pull the fabric taught around to the back of the chair seat and staple it tight. In order to get the best result, staple in the middle of one side of the chair seat, pull the fabric taught and staple directly on the opposite side. Continue all of the way around the chair.
Trim away excess fabric with scissors, leaving about an inch beyond the staple line.
Put the seat back in place on the cane-backed chair frame. Carefully turn the chair upside down, setting the seat down on a counter or tabletop if possible to keep the seat tight against the frame.
Screw the original screws back into the seat bottom. Trim fabric back from around the screw holes if necessary.