I don't know what the underneath material of the chair bottom is, wood? metal? fiber board? The idea of using a stretchy knit and pulling it tightly over the chair bottom either with elastic or a drawstring in a casing, is good because you will have the best chance of a smooth surface. But for longevity, you might conside buying a piece of faux leather (easily obtained in any fabric store or online), and smoothing it over the top of the chair bottom, turning the chair over and pulling the fabric as tightly and as smoothly as possible to the underside of the chair and either tacking it or stapling it if the chair botton is not metal. Duct tape is an option if the bottom is metal, and it wouldn't be a big deal to retape as needed.

I don't know what the underneath material of the chair bottom is, wood? metal? fiber board? The idea of using a stretchy knit and pulling it tightly over the chair bottom either with elastic or a drawstring in a casing, is good because you will have the best chance of a smooth surface. But for longevity, you might conside buying a piece of faux leather (easily obtained in any fabric store or online), and smoothing it over the top of the chair bottom, turning the chair over and pulling the fabric as tightly and as smoothly as possible to the underside of the chair and either tacking it or stapling it if the chair bottom is not metal. Duct tape is an option if the bottom is metal, and it wouldn't be a big deal to retape as needed.