Using your drill, remove the screws holding the seat and backing from the frame of the chair.
Remove the staples that hold the fabric in place with your pliers. Remove the old fabric and support foam; discard them.
Cover the seat and backing with new foam. Don't use an extremely soft foam, as sitting on a soft surface for long periods of time will result in discomfort for your client.
Cover the foam of the seat and backing in the new vinyl material. Stretch the material across the seat and secure it to the seat's underside with your staple gun. Place a staple every half an inch to ensure that the fabric is tight. Use a spill-resistant material. Expect that ink, blood and sweat will all make their way onto the fabric at some point.
Before reattaching the seat and backing to the chair frame, give it a thorough and rigourous cleaning. Most barber chairs are made of stainless steel, so purchasing a heavy-duty stainless steel cleaner is the best option. Apply the cleaner to all areas of the chair and rinse off according to the cleaner's instructions.
To remove any rust on the chair frame, rub vigorously with steel wool. Using a rust remover will work as well, but clean the area after use as most rust removers contain a corrosive agent.
Reattach the seat and backing with new screws and your drill. Do not use the old screws. They may contain rust, not conducive to maintaining a sterile environment for your clients.
Confirm that there are no loose parts and that the seat and back are secure against the frame.