I'd say an affine algebraic group over $R$ is an $R$-Hopf algebra, that is, a Hopf algebra object in the category of R-R bimodules. Further than that, it's hard for me to say.
[EDIT: This bit doesn't make any sense. Ignore it. I was up until 7am doing Mystery Hunt, so at least I have a good excuse.] I am pretty suspicious of a definition in terms of the functor of points; the whole problem with non-commutative geometry is that the points don't capture nearly enough information.