Set the chair in direct sunlight on a cloudless day. Allow it to sit or several hours. Sunlight has mild antimicrobial properties and can eliminate some smells.
Scrub the surface of the chair with the brush attachment on a canister vacuum while the vacuum is running. Move the brush in a circular motion. This will dislodge and remove surface dirt and dust which may be contributing to the odor.
Mix one-part vinegar with two-parts water. Pour the mixture into a spray bottle. Spray the seat of the chair with the solution until it is moist but not drenched. Allow the chair to sit for 30 minutes. Wipe the chair with a clean, wet cloth. Allow the chair to air-dry.
Mix one-part laundry detergent with four-parts warm water, using an electric mixer. Continue to mix until suds form and those suds peak. Dip a clean cloth into the suds and rub the surface of the chair. As you rub and as the suds do their work, dirt should begin to appear in the foam. Remove the foam, using a rubber scrapper. Continue working until the entire chair has been cleaned. Wipe the surface with a clean cloth and allow it to dry overnight.
Spray commercially produced foaming upholstery cleaner onto the chair. Let the foam sit for the time recommended by the manufacturer. Wipe off the chair with a clean, wet cloth. Allow the chair to air-dry.
Use the hose attachment for a carpet shampooer to deeply clean the cushion of the chair. Apply the cleaner and allow it to sit for 15 minutes. Remove the cleaning liquid with the shampooer hose. Empty the shampooer, refill it with clean water and repeat this process in order to remove any residual chemicals. Let the chair air-dry.