Honestly, I'd save up and invest in a high quality wig. It'll be worth it, trust me.
As for which dyeing method to use, it depends on the wig's fiber type (kanakelon, hiperlon, human hair, etc.).
ACen 2014 cosplay: Lara Croft, Erio Touwa, Shirahoshi Hime
ALA 2015 cosplay: Shirahoshi Hime, Akemi Homura [Haregi ver.], Makise Kurisu