Maybe start with dip dye and let your see they are not bad on your black hair (see from your pic) if not just say mum I am old enough to be able to make me own choices, do you want me to be a baby forever, i need to be independent and your not helping me
it is hard if she doesn't like, it is harmful if we get angry with each other, and nothing could be easier to be done in future. just try it again when she feels happier, and it might not be a good idea to do it by force.
In my opinion, your mom is looking out for you..so you won't be looking back at pictures from when your a teenager and being completely embarrassed as to WHY she even let you do that! That's how I am when I'm looking back at old pictures, asking my mom why she let me wear that or why she let me do my makeup like that..she just looking out for you!