Even as durable as leather is as an upholstery fabric, it's not impervious to hair dye. As one of the toughest stains that can occur on leather, the faster you work -- while the stain is still wet -- the better chance at success you have.
Blot up -- do not wipe -- the wet stain with a cloth. Continue to blot until the majority of the stain is removed. If the stain is dry, skip this step.
Test a small spot on the back of the leather with the store-bought leather cleaning solution of choice to ensure the leather doesn't fade or change color.
Spray the stain with an inexpensive can of hairspray that contains alcohol. Wipe the stained area with a clean, white cloth. The alcohol in the hairspray can remove the dye while it is still wet and hasn't had a chance to dry.
Dip a cotton ball in rubbing alcohol -- you can substitute vodka if you have it -- and blot the area of the wet stain with the alcohol. Replace dirtied cotton balls as necessary with a clean, alcohol saturated one until the stain is gone.
Let the area dry, and then buff the leather with a dry white towel or microfiber cloth until it shines.