Start with a quality makeup. Professional clown makeup is always good, as is theatrical makeup or professional makeup designed for modeling and photo shoots. Use a hypoallergenic makeup to avoid breakouts. Quality clown makeup and grease paint manufacturers include Ben Nye, Kryolan, Mehron and Paradise. Remember that well done clown makeup will cover most of the person's face, so using a good quality makeup is important.
The key to good clown makeup is to look at the person's features and accentuate those features to make them larger than life and comical. Before applying the color, use a white base on the areas of skin that will not have color applied. The white base will create more contrast and make the colors "pop" more. The white will also give the makeup more of a mask effect. In some cases, the person with clown makeup looks completely different than the person when not wearing the makeup.
Once the base has been roughed out, the colors that will accentuate the clown's natural features can be applied. You can make the outline of the lips cover a large percentage of the face and make big lips using black or blue makeup. Eyebrows can be made big, but should follow the natural shape of the eyes. However, if you've covered the eyebrows with white base, the eyebrow shape can follow the overall shape of the head. Eye shapes can either be mimicked or can follow the shape of the makeup eyebrows.