If you can get your hands on a copy of Cassels&Frohlich, /Algebraic Number Theory/, pages 106->107 essentially answer your first question (can they be developed axiomatically, and how to construct them explicitly). Also they do it for Tate cohomology of a finite group rather than plain group cohomology, but it should be essentially the same thing, since Tate cohomology is the equal to a group cohomology in degree >=1 (and equal to group homology in negative degree: just as you get group cohomology by taking the cohomology of the classifying space, you can also get group homology by taking the homology of the classifying space). (And anyhowe, if you want to learn local class field theory, you want the Tate cohomology version anyway.) There are probably nicer places to look for the explicit construction, but here's a summary of the axiomatic definition:

the cup product is a family of maps from H^p(G, A) \tensor H^p(G, B) to H^p(G, A \tensor B) for all A, B and all non-negative integers p, q (for Tate cohomology, put hats on all the H's and allow p, q to be arbitrary integers)

(i) these homomorphisms are functorial in A and B.

(ii) for p = q = 0 they are (induced by) the natural product A^G \otimes B^G -> (A \tensor B)^G

(iii and iv) they are compatible with the delta map in the long exact sequences associated to short exact sequences of G-modules in the following way: delta (a'' cup b) = (delta a'') cup b and a'' cup (delta b) = (-1)^(deg a'') a'' \cup delta b.

Writing out and checking the explicity construction can get annoying. In practice, though, at least when doing local class field theory, if you can come up with a map from H^p(G, A) \tensor H^p(G, B) to H^p(G, A \tensor B), it's probably cup product (okay, maybe up to sign).