How does one compute Chern numbers of spherical rational homology classes $$f: S ^{2k} \to BU.$$ These generate rational homology by MilnorMoore theorem since BU is a connected Hspace, and so $c_k$ cannot kill such a class. It seems very likely that $\langle c_k,[f] \rangle =1$ but what is the proof? Let me add here that I meant that $[f]$ is a MilnorMoore generator, i.e. it's the multiplicative identity in $\pi_{2k} (BU, \mathbb{Q}) \simeq {\mathbb {Q}}$.

We have that if $f\colon S^{2k}\to BU$ is an actual map of topological spaces (it is a little bit unclear from your formulation if you assume this) then $\langle c_k,[f]\rangle>$ is a multiple of $(k1)!$ and all multiple are possible. See for instance Husemoller: Fibre bundles, Cor 18.9.8, GTM 20, Springer Verlag. 

