Graham,
As Nap suggested, one of the techniques that will help you avoid the "zombie eye" phenomenon is to paint a thin line of black/ burnt umber along the underside of the eye lid to create a clear line separating the white of the iris and the flesh colour of the eyelid. Similar to women applying mascara, this thin black line helps to make the eyeball stand out and it also helps to close the shape of the eye to a more almond shape, rather than round.
I would also support the application of a very thin line along the bottom half of the eyeball from the inside corner of the eye to the middle of the eyeball. You can then apply a thin wash of red, as Nap suggests to soften the black line and create a slightly bloodshot effect.
My best advice is to look closely at your own eyes in the mirror and observe other people's eyes in different types of light to see how they look and the different ways they reflect and refract light.