Black hair dye, especially over-the-counter dye for home use, is very hard to remove. The color is dramatic, but the chemicals used to get the hair follicles so dark are saturated. They penetrate deeper into the hair shaft and hold the dark color. Removing black dye can be very damaging to your hair, especially if you want it out of your hair quickly. There are ways that will get the black dye out faster than normal, without damage.
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Wash your hair thoroughly with laundry detergent. Make sure you use a reliable brand of liquid detergent that's safe for colors, with no bleach in it.
Rinse your hair thoroughly. To make sure all the detergent is rinsed out of your hair, rinse it longer than you usually do.
Repeat Steps 1 and 2.
Condition your hair. After using harsh cleanser like laundry detergent, you must use a lot of the best conditioner you can on your hair, to keep it from drying out. You can also use a hot oil conditioner, if you want, because this will also help remove dye from your hair while conditioning it. Make sure to only use a hot oil treatment once a week.
Repeat this process daily to get the black dye out of your hair. It may take longer than a more harsh chemical treatment or bleach, but it will not damage your hair the way bleach or chemicals will.