An old swivel chair
Enough fabric to cover the back-rest and seat of your swivel chair
Step 1: Dis-Assemble your Swivel Chair
Both the back-rest and the seat of the swivel chair are probably screwed into the main frame, have a look at your swivel chair and dismantle it. On the one pictured there were two screws holding the backrest in place and four screws holding the seat in place.
Once you have unscrewed the back-rest and seat, you need to take the padded part out of the frame, with this chair it was simplya case of levering the padded part out with a flat-head screwdriver (used as the lever).