Before you can start washing, you need to remove makeup — or else the cleanser can't really reach your skin. If you're left with raccoon eyes after washing, you need to take the extra step of dissolving mascara and other eye makeup.
To remove eye makeup use a vegetable oil-based cleanser, such as Dermalogica's Precleanse ($34), or an eye makeup remover. These oil-based products liquefy stubborn liner and makeup without rubbing or tugging on the delicate eye skin. And with your mascara down the drain, your cleanser can really do its job on the rest of your face.
King offers up her technique for gently wiping away eye makeup: put cleanser on a damp cotton pad, lightly press it on the eye area and wait a few seconds, and then sweep the cotton pad down over the eyelid and eyelashes (this is also the best way to prevent makeup from getting into the eye). Repeat as needed, circling in towards nose but always sweeping down over the lashes. Open the eye, fold the cotton pad, and gently swipe under the eye. Rinse off the makeup remover to get rid of residue.