You can purchase an exfoliating scrub at your local drug store or make your own natural scrub using salt, sugar, or coffee grounds. Scrubs work by using small granules to remove dead cells from your skin, and leave your skin feeling smooth and healthy.  Dab a bit of the scrub onto your fingertips and gently massage it into your face using small, circular motions. Pay special attention to the area under your eyes, but take care not to apply too much pressure—the skin there is very delicate and can easily be damaged. Rinse your face with warm water and pat it dry. If you don’t like the feeling of the granules, try carefully dry brushing under your eyes with a long-bristled brush. Washing with a facial cleanser containing alpha hydroxy acid is a great option for those who find exfoliating too uncomfortable or irritating to the skin. The alpha hydroxy acid works to dissolve dead skin cells rather than sloughing them away like the scrubs.  Rub a squirt of the cleanser and warm water between your hands to create a lather. Gently massage the lather onto your face for about 1 minute. Try to spend some extra time on the area under your eyes, but be careful not to get the cleanser inside of them. Rinse your face with warm water and pat it dry with a clean towel. The ingredients in an anti-acne moisturizer dry out excess oil, which helps to unclog the pores under your eyes. This may be especially effective in treating whiteheads and blackheads.  Dab a small amount of the moisturizer onto your fingertips and gently massage it into the skin under your eyes. Take care not to get the moisturizer inside of your eyes. Use this moisturizer sparingly and only apply it to the affected area to avoid over-drying. Avoid using moisturizers under your eye area that contain salicylic acid because this may irritate your eyes.

Summary:
Exfoliate if you want to slough away dead skin cells under your eyes. Use an alpha hydroxy wash if you want to dissolve the dead skin cells. Apply anti-acne moisturizer under your eyes after removing dead cells.