Quick post from my iPhone!
To remove te backplate, rotate anti-clockwise. A suitable tool is required, but I MacGyvered using a Shimano bike-wrench. Creativity can bring you quite far! If all else fails, or you plan on doing watch-work frequently, consider a proper watch-opening tool
The grey plastic is snapped in place and can be lifted out easily with some care.
These small batteries are best bought online, or as part of a assorted collection (think dollar-store). The numbers on the cell actually indicate the proper type. I think this was a CR622, check your model carefully!