Easy, Illustrated Instructions on How to make giant Googlie Eyes for your Halloween Costume and Win Costume Contests page 2
Check how the pupil bowls and the eyeball bowls work together, then use hot glue to put them together.
Next, drill holes around the edge of the plastic bowl. Holes here will help with mounting the eyes onto your costume. Hot glue is probably strong enough to hold the bowl to the foam-core, but I add zip ties for strength.
You'll also have to remove the black handle to get the green plastic cover off. It is held on with four more hex screws. Take note of how the trigger is positioned within the two-part handle housing, because it will probably fall loose and have to be re-assembled after you have replaced the string.
Depending on the surface of your costume I recommend using glue or screws to mount the eyes.
The clips hold it in place, but allow it to spin in place. Loosen the screws for the clips enough to turn the clips out of the way. Remove the white spool.
If you are not satisfied with the look of the mixing-bowl eyes, try creating your own clear plastic domes by shaping your own hot acrylic sheets.