If you have oily or normal/combination skin, use a daily cleanser that has salicylic acid. Salicylic acid is an anti-inflammatory that helps remove dead skin cells. It controls oil production and increases hydration.
A cleanser with glycolic acid also helps get rid of dead skin cells. It's good for people who can't use prescription retinoid creams (like Retin-A and Renova). Read the label to see if your cleanser has these ingredients.
There are products specifically made to remove eye makeup, especially waterproof mascara. But some gentle facial cleansers can effectively remove eye and facial makeup, as well as cleanse the skin. Choose products that are fragrance-free to avoid irritating the eye area.