There is no simple way. Especially if this is a video with motion in it and ESPECIALLY if it was shot with a camera that wasn't locked down to a tripod.
If this is a still photo, you may be able to cut the eyeball out of the picture and place it on a separate video track so that it can be animated separately from the background.
But, as I say, if this is a handheld shot and there's motion in the shot, it would be very difficult to do it without an advanced special effects program like After Effects.
Actually, it is locked down. And I do have AE. I just wanted to make sure there wasn't something else I could do. But thanks for confirming.