Remove all the seams that attach the cover to the furniture with a seam ripper.
Flatten the fabric and use it as a pattern to cut the new cover. If the cover has any hems or folded edges leave them in place.
Fold 1 1/4-inch-wide acrylic bias tape in half and wrap it around the edge of the new cover. The edge of the fabric should be against the inside of the fold, and the width of the tape should be the same on both sides of the fabric. Hold the tape in place with straight pins. When you have pinned the tape all the way around the fabric, fold under the last 1/2 inch on the end of the tape. Lap the final end over the first end 1/2 inch.
Install a denim needle in a sewing machine and thread the machine with nylon upholstery thread. Set the machine to sew a straight stitch with a standard stitch length. Sew 1/2 inch from the folded edge of the bias tape to hold it in place. Remove the pins as you sew. The bias tape will finish the edges of the new cover.
Thread an upholstery needle with 36 inches of nylon upholstery thread and knot the ends of the thread together. Replace the cover on the chair and hand sew the seams. When you reach the end of the seam, sew back to your starting point, going up through the holes you went down through and down through the holes you went up through. The stitches of the second seam will fill the spaces between the stitches of the first seam. This technique will strengthen the seam and improve its appearance. Knot the thread close to the fabric and cut off any excess.