First up, remove the rear air vent panel. It's tempting to just remove the two silver air vents themselves, but it's quite hard to do that without breaking the interference washers that smooth the adjustment of the air vents (ask me how I know). So just pry the whole unit out. It's press-fit; there aren't any clips.
Remove the two screws that were behind the air vent panel - try your best to not drop them down the HVAC ducts because that's probably the last time you'll see them.
Lift out the trim cover from the back of the center console unit (i.e. towards the front of the car). Inside it is some ducting that just rests in place.
Remove the wooden trim piece that surrounds the screen. It's held on only by two metal clips at the top - you can squeeze the clips with pliers and then push them through from the other side.
Remove the two 12mm brass bolts under the rear of the screen - these hold the tray table. Don't drop those down the HVAC either. I don't have a very good pic of those but they're roughly here:
At this point the tray will slide out. It has a curved plastic piece behind it - this just rests in there between the frame of the table and the bracket. It only fits one way up (where some little flat tabs meet the wooden screen surround).
Reinstallation is the reverse of removal - just remember to put the tray in before the wooden screen surround, because otherwise it's not very easy.