Yes, bleaching your whole head is how to get a black hair color out of your hair quickly. You can try other methods, because bleaching hair can be really damaging, but it’s a quick fix.
Go to Sally Beauty Supply or a local beauty supply store and buy bleach powder and some 20 or 30 volume bleach (I’d suggest), and you should take your time with lightening your hair. I’d say that you should lighten it once every two or three weeks and make sure you’re deep conditioning it in between to prepare for it to take the damage. It might not be fun and you might get some really unattractive colors out of it but it will be worth it in the long run.