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