It seems to me that the first thing to do is to define the signature. It is a very important morphism, and you need to have a bunch of example of morphisms to present to your students anyway. Only then is the alternating group really relevant, and you get all properties you want from what you will have done with the signature.
Of course, I am telling you to pass all the difficulties from the introduction of the alternating group to the introduction of the signature. But it seems that this way is much more natural, hence should be easier to convey convincingly.
For example, following the comment of PVAL, you can first show the important property that every permutation is a product transpositions, and ask whether the needed number of transposition is well defined. Of course it is not, point this out. Then what one could try is to associate to each permutation the minimal number of transposition needed to write it, but then it easily turns out that this map does not define a morphism (but it is linked to a lot of interesting math, from geometry of groups to work of Helfgott and others on the diameter of finite groups far arbitrary set of generators). Then, when you show the students that looking at the parity of the number of transpositions generating an element yields a morphism, you are in a very meaningful context.
There are other ways to introduce signature, for example looking at the number of pairs $(i,j)$ with $i<j$ whose order is changed by the permutation. The above one seems best to me right now, but if anyone can suggest other in comments I would be happy to have more choice when I have to teach that myself.