Arduino powered servos control the movement ... There are plenty of approaches to Animatronic eyes. Check out the eyes in this Stargate Horus helmet, which just won our Sci-Fi contest.
The eye dimming and closing, combined with the head movements, make it look almost alive. This isn’t [jerome]’s first Stargate-Horus-Helmet rodeo. He made an Arduino-powered one back in 2011.