Prof. Conrad mentioned in a recent answer that most of the (introductory?) books on reductive groups do not make use of scheme theory. Do any books using scheme theory actually exist? Further, are there any books that use the functor of points approach? Demazure-Gabriel's second book would have covered general group schemes in this way, but it was never written, and it's not clear whether or not it would have covered reductive groups anyway. There is a lot of material in SGA 3 using more modern machinery to study group schemes,
but I'm not aware of any significant treatment of reductive groups in that book (although I haven't read very much of it).
Correction: Prof. Conrad has noted that SGA 3 does contain a significant treatment of reductive groups using modern machinery.