Purchase real grease paint. You'll want black or a dark gray and white. Avoid using simple make-up; it's not going to do the job.
Start with paling the facial skin. Apply white grease paint all over the face. This'll make the fake rings under your eyes stand out. Some experts recommend mixing a little gray into the white make-up or even some yellow or green to make your face a really disgusting color.
Shine a flashlight onto your face so your eyes are shadowed.
Paint in the shadows cast from the flashlight using your black or dark gray grease paint.
Use your finger or a make-up sponge to smear the grease paint so it looks more realistic.
Add a faint dusting of baby powder or cornstarch over the skin to help set the grease paint and prevent you from sweating it off or from leaving patches on your costume.