Turn the dough out onto a clean counter or silicone mat, and knead vigorously until it becomes silky-smooth. Divide the dough into balls for coloring.
Make a divot in the center of the ball, and drop some food coloring1 in. Fold the dough over, working the food color through the body of the playdough, trying to keep the raw dye away from your hands and the counter. You could use gloves, a big ziplock bag, or plastic wrap at this stage to keep your hands clean- only the concentrated dye will color your skin, so as soon as it's worked in bare hands are fine.
Work the dye through, adding more as necessary to achieve your chosen color.
1 If you use Kool-Aid or similar unsweetened drink mix for color, test on a small ball first- it won't go as far as the "real" food coloring.