You call changing your hair color "coloring your hair".

You also can say that someone "dyed her hair", but "dye" doesn't sound as nice. It brings to mind the image of the actual dyes (which are chemicals which change the color of hair or fabric). And you can "dye" hair, clothes, and Easter eggs. But you can only "color" hair.

So use "color" to talk about someone's new hair color and how it looks, and "dye" if you're talking about the actual process that they have to use to change the color.