Flip your chair over so that it is upside-down with the legs in the air. If the chair has a back, place the flipped chair on the edge of a counter or table so that it can sit flat.
Unscrew the seat cushion from the chair's frame and set the screws aside.
Pull the staples out of the cushion. The staples are what hold the current upholstery on -- get under them with a staple remover or claw hammer and yank them out.
Lay the old cushion cover on a piece of foam batting. Cut a circle out of the batting that is about 1/4 inch wider than the seat of the old cover.
Lay the old cushion cover on your new fabric, and press it completely flat to give yourself a pattern. Trace around it on your new fabric and cut it out.
Layer the new piece of foam batting on top of the exposed cushion for extra padding. Wrap the new fabric around the cushion, pulling the edges around to the bottom like the old fabric was. Pull the fabric tight around the cushion, and staple it to the bottom of the cushion all the way around.
Screw the cushion back onto the chair's frame.