Use a hot-glue gun to adhere padding onto seat and down the sides. Hold in place while the glue dries. To make it extra cushy, glue a second piece of carpet padding on top of the first one.
Cut a piece of quilting batting bigger than the carpet padding and lay it on top. Use a hot-glue gun again to adhere the batting to the padding.
Use cotton batting for the final layer. Cut this layer long so it hangs down past the bottom of the seat. Glue cotton batting onto the quilting batting on top and around the sides of the seat. After the glue has dried (usually just a few minutes), flip the chair over and staple the overhanging batting to the bottom of the chair seat.
Again, use the old fabric as the template to cut the new fabric. Place the new fabric on the seat and make sure it is centered and smooth. You may want to stick a few straight pins in the top to hold the fabric in place. Place one staple in each side, in the middle. Turn the chair upside down. Pull the fabric a little taunt and continue stapling each side from the middle out to the corners. If needed, use a hammer to push the staples all the way down.