Remove existing fabric. This can be done by ripping it off or carefully removing any screws, staples or tacks that hold it in place. In the case of a mostly wood chair, you may be able to completely remove the seat and also possibly the upholstered part of the back, and then the fabric can be easily removed.
Inspect and assess the quality of the padding. This may be able to stay, or it may need to be replaced. Remove it if it needs to be replaced, and replace it with batting or foam padding, which you can glue or staple into place. The fabric will help hold this once it has been attached.
Cut the fabric to size. If you have a simple chair that only has an upholstered seat, measure the seat and allow about 6 inches on all sides beyond the width and length of the seat itself. If your chair is mostly upholstered, you may need a pattern. Depending on the condition of the fabric you removed, you may be able to use that as a rough pattern. The link Upholstery Techniques: Upholster a Chair in the References section has a sample pattern as well.
Attach the fabric in place with staples, stretching it taut as you work. For the seat, start on opposite sides and do the centers of each straight side first. Leave the area within 4 inches of the corners till last. Hold the extra fabric up where it meets at the corner and cut diagonally toward the corner, still leaving some material to overlap. Then fold it so it gathers in diagonal lines from the corner pointing toward the center of the underside of the seat, and staple each corner several times. If you have a removable upholstered back area, you can cover it in much the same way, though there will be more gathers if it is an oval instead of a square.
For a mostly upholstered chair, the arms and back will be covered separately, and you will probably need to cut a separate piece for the back of the chair, and also maybe for the outside parts of the arms. Again, use the original fabric as a guide since every chair may be a little different. For the main part of the back, you can pull the fabric through between the back and the wings and staple it to the back. Folding the fabric over the arms is a bit trickier because there are likely to be gathers, but it works in a similar way. The fabric that covers the inside and edges of the wings works similarly.
Attach the back and side panels. These panels are where you hide all of the staples that show on the wings, the top of the back and the sides of the arms. You do this by stapling welting cord along the edges, and attaching flexible jaw stripping next to it. Attach the outside panel and close the jaws as you go, pushing the fabric edge into the jaw with a sharp object.