Because of the condition of the adhesive for the screen assembly, I decided to replace the adhesive completely. You could skip this step, but I figured it was worth cleaning it up.

To remove the existing adhesive I just used the sharp razor blade and worked my way around peeling it all off.

NOTE: This would be a good opportunity to 'test fit' the new screen (without adhesive) so you can see where the cables need to go. There is one which goes up behind the screen and through the middle of the phone, and another which goes through the top hole. The top one is a strange shape, and needs to go through sideways first.