Can anyone refer me to a proof of uniqueness of the cohomology of CW complexes, as follows?
There is a natural isomorphism$$H^*(X, A; \pi) \cong H^*(C^*(X, A;\pi))$$under which the natural transformation $\delta$ agrees with the natural transformation induced by the connecting homomorphisms associated to the short exact sequences$$0 \to C^*(X, A; \pi) \to C^*(X; \pi) \to C^*(A; \pi) \to 0.$$
May's A Concise Course in Algebraic Topology mentions this result but does not supply a proof.
You should be able to find a proof in most books covering homology theory. A specific reference is Theorem 4.59 of Hatcher's Algebraic Topology (p. 399-402). As Mike Miller commented, the idea of the proof is to show that any (co)homology theory can be computed by a "cellular chain complex" associated to that theory, and that these cellular chain complexes for different theories are all naturally isomorphic to each other.