How do I incorporate cucumbers into cole slaw?
If you want to match the texture of the shredded cabbage you could use a peeler to get thin, long strips of the cucumber.  If you want to contrast the texture you could cut it into chunks or slices.

I would discard the seed pulp to cut down on the amount of water that will fall to the bottom of the bowl.

The skin is completely personal preference.  If it's thin and tender I like to leave at least strips of it on for color contrast.  Otherwise I get rid of it to avoid the tough, chewiness it can bring.