Rather than trying to position plastic eyes correctly, I've been making needle felted eyes for my more recent stuffed creatures. I tried finding a tutorial for someone recently but didn't see one, so decided to make my own since I was about to needle felt the eyes onto the purple mystery pig.
Step 1: Gather Supplies. In this case, it's the object to needle felt onto and some lengths of scrap wool to felt. The object can be any sort of fiber but the felting scraps should be wool. Make your scraps fairly generous for the area you want to cover, you can almost always felt it in tighter and smaller.
Step 2: Starting Point. Begin to tack down the center color (pupil, whole eye, whatever you're doing) where you want it.
Step 3: Coil Around. Wind the yarn around the starting point and felt it down. Keep winding and felting to cover the area you'd like covered.
Step 4: Next Color. Do the same steps of tacking down the beginning, then coiling the yarn where you want it and needle felting it in for any additional colors you're using (iris, white, eyelashes, etc.).
You can make the areas as fluffy or as tight as you want by tacking down only a little bit along the yarn scraps or fully felting it down. The eye on the left is lightly tacked down while the one on the right has been more vigorously felted.
One of the nice things about starting with yarn scraps is that you can more easily make straight lines with the yarn where you want it to be. As long as the yarn scraps are wool or something that will felt, you're good to go! You can also do fun things like felt down one side of a piece of yarn and brush out the plies to make eyelashes or other hair-like things.