Parent the eyes to the bones in the head.

Why are all the characters we create bone-heads?

I ran into a problem with parenting one time while writing code to create a skeleton automatically and after a bit of frustration with it, I added 2 joints to the skeleton and parented the eyes to them . . . . after you skin the character though, you have to select the skin, then select the eye joint, and go to Skin>Edit Smooth Skin> Remove Influence. That stops the joint from having any effect on the skin. Unconventional, but effective.
I suppose you could leave the influence on the skin and just rotate the eyes, but I had coded the joints so they could control the eyes with sliders . . . . hee hee haa haa ho ho I gotta be the true blue big bull looney in this forum . . . :attn: