i want to dye my hair blue, red and purple?

Dye it with the est you can afford. Professional is best. Wash it less and it will last twice as long. Dry shampoo is an option. Also, use color protecting shampoo and conditioner.

Color Briliance

Nothing as hair grows at a fixed rate and your natural roots will show though always at the same growth rate. What you can do is to pick a contrasting colour knowing that your natural colour will grow back and use that growth to give yourself an ever changing, dynamic hair colour mix. So if your hair was normally black dying it white you would start with all white hair and slowly you would get black & white and eventually just white tips and then back to black again once the whit has all grown out. (assuming you get you hair cut occasionally).