How can I cover my desk chair's seat?
You should be able to turn the chair over and unscrew the seat part. If the cover doesn't come off (probable), you'll need to trim the loose pieces and make patterns for each section. If you don't want wrinkles, you need to cut fabric out in the pattern of the original seat. You need to allow for seams and fold over the edge (measure the edge into the plate that screws the seat to the chair. An easier option without the pieces is to use a stretchy woven or a firm knit, but in time they will sag.
You can probably just use duct tape to attach the fabric under the seat plate.