The eyes were easy.
And they shouldn't be. The eyes are the window to the soul, as they say, and that is where the audience will spend most its time looking. To that effect, eyes deserve all sorts of wonderful deformers to allow them the greatest range of expression possible. For much the same reason for other parts of the rig, this simply can't be done with a game engine (yet).
But we can cheat it very nicely :) My approach to the eyes was as straightforward as they come. I plopped a joint right smack in the middle of the eye, and bound the eye to that. The eye rotates along with the joint at its center, and all is right with the world.
What about eyelids? Well, for those I just duplicated the eyeball joint twice. One for the top and one for the bottom eyelids. Then I painted the weights of the joints on the skin around the eyes veeeeeery slowly, testing every few strokes to make sure I had a good balance. I took a screenshot of my weightings to give an example. The weight display is inverted and clamped to be easily visible! In actuality, the weighting appears very muted and gray because of how I spread it out around the eye.
You'll notice up above I have a third image as well. To add just a touch more believability, I shared a little bit of the skins weighting with the original eyeball joint. That way, as the eye looks around, it sort of snags the skin around it. It's not much, but it does add a bit of life to the character.