I've done lots of dye work on animal furs, so I can give you a few pointers here. Do you want to tan a whole hide, of just part? Are you going to mount it or use the hide for making something? Regardless of your final use, you will want to work on a hide that has been dry-tanned. That will get you best results and virtually no risk of damaging the hide (slipping, etc).
Your best bet will be to use permanent (or semi-permanent) hair dye that you can buy from a beauty supply store like Sally if you have one in your area. They usually have all sorts of colors; magenta is pretty easy to find. The ones I like come in a plastic jar (the ones in the tube don't seem to grab as well) with a black lid. There are various shades of pinks, purples and everything in between. To take a fox from red to magenta isn't too hard. But if you want it "flaming" bright you will likely need to bleach the fur a bit before you apply the dye. You can do that with creme peroxide and whitening powder - the same stuff people often use to whiten skulls. You won't need to make it white (which is hard, you'd have to bleach at least twice for that and it's hard on the fur), just lighten it a bit.
If you put the dye on an unbleached hide you'll get a darker magenta, on a bleached one you'll get a brighter color. But if you are going to do a whole hide you will need probably two jars of the stuff, at least.
Here are some pics of a fox I put some dark pink dye on (without bleaching). The black is just black hair dye painted on:



To get a stronger color than that you will likely need to pre-bleach.
But basically that's what you do - just follow the mixing directions on the dye, but you will probably want to leave it on longer than recommended.