Blot the spot with paper towels. Don't rub or scrub -- this spreads the color around and can leave a bigger mess. Soak up the mess using paper towels until you can't get any more color off of the wall.
Mix 1/4 teaspoon dish-washing soap with a cup of lukewarm water. Dab the mixture onto the wall and let it sit for a minute or two.
Blot the area with paper towels again. The idea is to saturate the stain with warm water and soap to loosen the dye so you can blot up more color. The bigger, deeper and older the stain, the more applications of soap and water you're going to have to use.
Let the area dry and go over it with a microfiber sponge. Microfiber sponges are heavy duty scrubbers that do little damage to walls.
Moisten a cotton ball with hydrogen peroxide and swipe it over the area. If you were going to dye your hair lighter, you'd have to bleach it first. The same idea applies to your walls -- hydrogen peroxide bleaches whatever stain is left when soap and water won't get the job done.
Let the hydrogen peroxide dry and apply again, if necessary.
Opt for rubbing alcohol on marble or stained surfaces, which aren't as porous as wood or painted walls. Rubbing alcohol is harsh, but makes for quick cleanup of black hair dye. If you don't have any rubbing alcohol on hand, spritz the area with hair spray. Most low-end hair sprays contain some alcohol, giving the same effect.
Don't give up -- it can take time and persistence to get black hair dye off of walls. If it takes a week or more, bite the bullet and consider repainting the area.
Don't use hydrogen peroxide if you have medium or dark colored walls. Since hydrogen peroxide is a bleaching agent, it's best used on light or white walls.