First of all cool idea for a model, the face thing works quite well

As for the eyes, the best way to do it wud be to unbind the skin, then seperate the eyes, possibly the eyelids as well, from the rest of the geometry.

Then rebind ur skin to the skeleton, excluding the eye geometry. This will allow ur guy to bend but he'll leave the eyes behind. So to fix that problem, don't bind the eyes to the skeleton, just parent them to the head joint. Do this by selecting the eyes first, then the head joint, then press p. The eyes will now follow the rotation of the head.

One last problem u mite have is the head deforms away from the eyes. To fix this just make sure that the skin weights for the top of the head, particularly around the eyes, is assigned only to the head joint. (weighting of one). Now the head shud only deform with head joint.

Hope this helps.

And to take a shot of ur work view u need some kind of screen capture software. just do a search in google, theres plenty of free ones.