The most important thing behind painting eyes is steady hands and thin paint.
First of all, paint with your hands together, because everybody has a certain amount of shakiness in his hands, but by painting with hands together you can clear the shaking out mostly. But still, experiment first on seeing how to reduce your shaking as much as possible.
Light is another important factor, you might use a lamp that you can aim directly above the eyes, so you get a good view of what you are going to do.
Take out white paint (or whatever colour you want to use for the eye) and thin it down to where it’s just not clumpy anymore, so it will go on smoothly. Then paint the eyes carefully and try to stay within the lines of the eye. If you do accidentally spill some, make sure to repaint only when that colour is dry.
For the pupils in the eye, there is just not really a trick to, just focus and know where you want to put it. Thin down your paint again to where it’s just not clumpy anymore, all in the sake of smoothness. Now take out your smallest brush (if you haven’t done so already). I usually use a “Games Workshop, Citadel Fine Detail Brush” for this. Aim at your invisible target, and very slowly move your brush toward the eye, so you can re-aim if your not heading to where you want to (for example his nose, well actually if you are that far off you will not be able to paint the eyes anyway). Once you’ve placed the dot of black (or any other pupil colour) in the correct position, you are done with the eyes.
I like to paint my eyes at the very end of my model, when all my painting is done. This gives me some sort of freedom, to decide whether I even WANT to paint eyes or not. Sometimes a model looks cooler without them.
Hold your hands firmly together and experiment with how to reduce your hand’s shakiness. Also try to paint your model’s eyes at the very beginning of the face, that so, if you make any mistakes you can cover that up easily later on. If you hate eyes anyway, experiment with painting them closed, because on most LOTR models this is very well possible. If you have trouble with painting eyes, try painting 40K Space Marines, they have eyes that should be fairly simple, and with some washes you can actually achieve some great effects on those eyes.
As an example, try painting 40K SM eyes white, and I mean totally white eyes. Then take out blue ink and thin it down with water, now you’ve got a wash. Put the wash on your model’s eyes and wait till it’s dry. Now repaint the centre of the eye with white (again thinned out till it’s not clumpy anymore). You should now have eyes that look simply amazing.