I'm not sure what your current sources are, but the definitions are laid out clearly in SGA3 (by Demazure and Grothendieck) and similarly in the book by Demazure and Gabriel, *Groupes algebriques* (North-Holland, 1970) which was later published in an English translation.   (Their designation of this book as "Tome I" is of course unfortunate, since it had no sequel.)

In Demazure-Gabriel, one finds for example an explicit statement about the existence (over a field) of a faithful linear representation, in the affine case:  see II, 5.2.    This is far into their book but is fairly elementary, just relying on the basic notions.

For a treatment heavily influenced by Demazure-Gabriel (or SGA3), you can also consult the early sections of Jantzen's book *Representations of Algebraic Groups* (Academic Press, 1987; 2nd enlarged edition, Amer. Math. Soc., 2003).    See especially I.2 for the notion of rational representation of a group scheme (over any commutative ring).

I should add that Jim Milnes has developed a modern textbook version of all this, probably published by now; check his webpage for details: http://www.jmilne.org/math/