This is a special case of the hyper(co)homology spectral sequence from Chapter XVII, Section 2 of Cartan-Eilenberg (1953), for the functor $T(C,A) = Hom_R(A,C)$.  The $E_2$-term is given in equation (4) on page 368, essentially as
$$
E_2^{p,q} = \prod_{s+t=q} Ext^p_R(H_s(A_*), H^t(C^*))
\Longrightarrow H^{p+q}(Hom_R(A_*, C^*)) \,,
$$
with $p$ as the filtration degree.