The programme that I propose for answering questions of this kind is described in my paper Foundations for Computable Topology. I had in mind that this might be done for (the opposite of) the category of (commutative) rings, though I confess that my efforts to do so drew a blank.
Thank you for telling me about Schlomiuk's paper. To me now it seems pedestrian and too closely wedded to sets of points, but that is completely unfair because it was written a long time ago. (I would say that it resembles Lawvere's work on the category of sets long before the invention of elementary toposes.)
I notice that Schlomiuk includes an axiom with some similarity to what I call the Phoa Principle, in the form that the Sierpinski space (which he calls $E$ and I call $\Sigma$) has just three endofunctions. It also uses extremal monos, as I have done, coincidentally with the same name but in ignorance of previous work.
The analogue of the Phoa Principle for affine varieties would be that any endofunction of the base ring qua space is a polynomial.
The analogue of my exponential $\Sigma^X$ for rings should be polynomial ring or free symmetric algebra on the underlying abelian group (ie forget the existing multiplication and freely adjoin a new one). However, whilst this works fine for locally compact frames, for rings it goes badly wrong for cardinality reasons.
There is some work by Mike Barr on Banach algebras that might help here, though I couldn't get my ideas to work with it; I will look out the references if you are interested.