Here's the chair before I started:
The first thing you do is unscrew the seat from the chair. Easy, peasy.
Then you remove the staples (I sort of just pull the fabric up and it rips the staples out) and discard te old fabric. Next you lay down (obviously, face down) the new fabric you want.
Then you just work your way around the seat, stapling (with a staple gun) and gathering the fabric so it's cinched and taut all the way around.
Trim the excess fabric off (it will be covering the holes you need to access to screw the seat back in) then just screw the seat back in.
Nothing to it! Isn't it pretty? As with every home project after I've finished it, I'm asking myself why I didn't do it sooner!