Carefully rip up any folded edges of the fabric that were glued down. Don't rip the fabric itself, just tear the folds up so that you can access the tacks and seams underneath. If the fabric is glued to the chair anywhere, use a glue thinner to help release the hold of the adhesive.
Pull out any decorative upholstery tacks if there are any lining the edges of the chair.
Look carefully at any seams in the fabric. If the covering is truly stitched together, use a seam ripper to pull the stitches out. Don't simply start ripping or cutting the fabric.
Use a tack remover or needle-nose pliers to pull out any upholstery tacks holding the fabric onto the chair.
Pull the fabric off. Remove any foam underneath the fabric if you are putting new foam on as well as new upholstery, or if the old foam appears sagging or rotted.
Avoid pulling out or moving any stuffing or batting, including cushion casings. Unless they've been destroyed for some reason, you are going to be recovering them. Just smooth or reshape any bumps or lumps.