You really shouldnt grow out grey hair, the best thing to do is find a hair colourist in your area,
If your hair has been permanently colored or bleached, a short haircut will be the quickest. Then you can use a semi-permanent hair color or hair color rinse in your natural shade to blend till your hair grows out. You can have a hair colorist blend your new growth with “lowlights”. Lowlighting is just the opposite of highlighting. Depositing streaks of hair color using a semi-permanent hair color to blend grey hair until the old hair color has been cut out. Your grey hair will start to come in naturally but without a line of demarcation.
The darker your hair color is, the trickier the process becomes. The process of lowlighting hair may take a bit longer. This is why going with a short haircut would be a good idea. It will make the process much smoother and quicker.