Avoid synthetic hair colors if possible. They’re made with harsh chemicals that can put a nasty burn on sensitive skin (and facial skin is more sensitive than scalp skin).
Use Black Henna instead; I recommend Light Mountain brand in the square box with the picture of a toucan. Black Henna is a blend of regular botanical Henna powder and powdered Indigo; Light Mountain is the best one for beginners IME, because it’s the most convenient to prep and use.
One four-ounce packet of LM Henna is enough to color a whole head of long hair, so a quarter of that would probably do for a goatee. Other than that, just follow the easy instructions from the package. If you want your beard to be really black, keep the Henna for the full suggested duration before you rinse it out.