What structure does the alternating group preserve?

The alternating group preserves orientation, more or less by definition. I guess you can take $C$ to be the category of simplices together with an orientation. I.e., the objects of $C$ are affinely independent sets of points in some $\mathbb R^n$ together with an orientation and the morphisms are affine transformations taking the vertices of one simplex to the vertices of another. Of course this is cheating since if you actually try to define orientation you'll probably wind up with something like "coset of the alternating group" as the definition. On the other hand, some people find orientations of simplices to be a geometric concept, so this might conceivably be reasonable to you.

$A_n$ is the symmetry group of the chamber of the Tits building of $\mathbb{P}GL_n$. The shape of this chamber is independent of what coefficients you insert into the group scheme $\mathbb{P}GL_n$, just the number and configuration of chambers changes. If you insert the finite fields $\mathbb{F}_p$ then you get finite simplicial complexes as buildings, and the smaller $p$ gets, the fewer chambers you have. You can analyse and even reconstruct the group in terms of its action on this building. The natural limit case would be just having one chamber and the symmetry group of this chamber - the Weyl group - is $A_n$. This is how Tits first thought that there is a limit case to the sequence of finite fields - which he called the field with one element.

Maybe somewhat more algebraically you can think in terms of Lie algebras - as I said the shape of the chamber does not change with different coefficients. The reason is that it is determined just by the Lie algebra of the group and thus describable by a Dynkin diagram or by a root system (ok, geometry creeps in again). The Wikipedia page about Weyl groups tells you that the Weyl group of the Lie algebra $sl_n$ is $S_n$. If have no experience with Lie algebras, but maybe you can get $A_n$ the same way.

If you can get hold of it, you can read Tits' original account, it's nice to read (but geometric) see the reference on this Wikipedia page.

Edit: Aha, I found a link now: Lieven Le Bruyn's F_un is back online. You can look there under "papers" and find Tits' article. And, since you are picking up the determinant ideas, you should definitely take a look at Kapranov/Smirnov!

Here is one idea, although I do not find it very satisfying. An object of $C$ is a finite set $X$ equipped with $\frac{|X|!}{2}$ (or $1$ if $|X| = 1$) total orders, all of which are even with respect to each other (in other words, basically a coset of $A_n$ in $S_n$). A morphism between two objects in $X$ is a map of sets preserving these orders (in other words, take one of the orderings on $X$ and apply a function $f : X \to Y$ to its elements. The result, after throwing out repeats, must be compatible with an ordering on $Y$.)

This is more or less a discretization of Omar's answer. Again, I would like to do better than this, or at least see the data described above packaged in a more satisfying way.