Whether you like your cheesesteak sandwiches "wit" or "wit-out" onions, the secret to slicing rib-eye, or any other cut of beef, is in the temperature of the meat. Putting it in the freezer before slicing it makes it firmer and therefore easier to cut.
Wrap the rib-eye in plastic wrap and place it in the freezer for about 20 minutes, or until it's firm to the touch, but not frozen solid. Lay the steak on your cutting board and slice it thinly against the grain. Make sure your knife is sharp; you're more likely to injure yourself with a dull knife. Keep the fingers of the hand that's holding the meat curled back toward your palm. That way, you won't cut your fingertips as you slice.
Chill the meat in the freezer; then use your food processor to slice the meat. Insert the slicing blade and cut the meat in chunks that will fit in the feed tube. Exert uniform pressure as you slice the meat so the pieces are all the same size. If your food processor doesn't have slicing blades in different sizes, experiment with an uncooked potato until you're confident about the amount of pressure you have to exert to get the right thinness. If you have a meat slicer, you can easily determine the thinness of the slices.