It wasn't necessary to remove the backrest entirely, which was a good thing because we couldn't fathom how to do so. Instead, we unscrewed the plastic sleeve that covered the seat height mechanism, hoiked out dozens of staples along the bottom edge and the snipped carefully around any fabric still covered by the secondary plate beneath the sleeve. Then it just peeled off. Putting on the new back was done in reverse, with the fabric stretched and stapled in place along the under edge, turning it under where it abutted the secondary plate of the height adjustment assembly and then extensions to the piping casing were overlaid over those raw edges, with their raw edges turned under and stapled in place also. Once the main plastic sleeve was screwed back in place, any remaining raw bits at the ends of the piping casings were covered up.
For the seat, we removed the arms (in this model they're held on with an adjustable screw so easy to do), turned it upside down on the picnic bench outside and removed about five or six screws holding it to the steel base. At first we couldn't understand why it didn't just lift off, but in this model there are hooks in the steel base that engage with the underside of the wood of the seat but a firm thump to push it backwards released it. Then it was a case of hoiking out dozens more staples.
To reassemble we reversed the process, stretching the new cover over and stapling it down, being careful not to lose the screw holes - we stuck old staples in them temporarily. Once we'd finished stapling it down, I used my sharp needlework scissors to snip gaps in the fabric over the screw holes and assembly was done in reverse of disassembly.
Top tip, though - a power screwdriver with hex and cross bits made this a lot easier than it might otherwise have been - I couldn't get the screws to move using a hand-driver.
The plastic raised edges might make things easier - fewer pattern pieces will be needed, I expect. But you will need to investigate how the seat cushion is attached to the base. If you can remove the arm rests, it makes it easier to turn it upside down on a table and you should then be able to see any screws.
I would say that this kind of thing is a two-handed job - the need to stretch fabric while operating a staple gun on narrow areas while preventing the chair from moving means it's a lot easier if two of you are doing it.