All you really have to do is:
* Desolder the original battery
* Solder in the new battery
* As for polarity, ground/black is the upper pad, +3.7v/red is the lower pad, but double check before you desolder the original.
* The GCW will immediately power back on as soon as both leads have been connected.
* Use whatever lithium-polymer battery fits, really. I think the rough dimensions of the stock battery (by the model number printed on it) are 4.6/36/108mm (depth/width/length). There may be more headroom in length and depth than width, however.
* Ideally, you'd want to splice in a JST connector; much, much less can go wrong.