First, I found some good sturdy fabric. I was given a ton of some strawberry quilted material years ago and had quite a bit left, so I decided to use it.
I really looked over my chair's seat part and decided it might be best to sew this in two parts-one part being the back and the other being the actual seat...these would connect about where your 'bottom' would hit the chair. I used that as a guide-making certain to allow for the seam to connect those two pieces.
This is the back of the chair-you will not need to worry about the grommet part of the back since you are going to just slide your new seat cover over this existing piece of canvas. Lay your existing seat over your fabric and cut out the 'bottom' part of your new seat. Allow for seam allowances and the small pocket at the front of the seat for slipping over the front of the chair-more on that later. Then, flip your existing seat around and do the same for the top part. When you are cutting the top part, allow quite a bit extra for making the pocket that slips over the seat. I cannot tell you how much, because you will really want to try the fabric on your chair, see how much is needed to fold over but still not interfere with any bars (this is mostly important on the front of the seat part :) Basically, be ready to use your chair as a sewing dummy :)
I sewed the seat part and the back part together. With this sturdy material, I just made a really sturdy seam by sewing it twice-once on a close zigzag and another on a close straight stitch. Now, for my other daughter, I used a material that was not quite as sturdy, so I sewed hers differently. After connecting the top and bottom pieces, I folded over the seam at the 'tushie' area and sewed it down to the seat part. I actually have no picture of this, so if this is confusing, ask me and I will try to 'create' a picture :)