Remove the original fabric from the chair. I did this by pulling at the edges with a pair of needle nose pliers.
Next, I added a little extra cushion to the seat and stapled the fleece (or batting) a little higher than where I was planning to staple the new fabric. Pull the fleece or batting tight before stapling.
Using the original fabric pieces as patterns, cut new pieces from my vintage ticking stripe fabric.
Take into consideration how the original pieces were attached. I stapled the new fabric on pulling the fabric tightly so it fits snug.
Finally, I hot glued the trim on in all the places it originally was.
Notes
Antique chairs are usually worn, and less comfortable than they once were. To remedy this, add batting to the seat of the chair.
Remove the old fabric panels as carefully as possible so you can reuse the pieces of fabric as a pattern for the new fabric.
Be careful when removing the old fabric, they usually contain upholstery tacks or tack strips. They can hurt.
Look for fabric at antique stores, sale sections at the fabric store, or Amazon for the best deals.