I've asked a version of this question on MathOverflow, so here's a hyperlink for continuity. If I get an answer on one, I can CW the other (or, you can answer both )
Let $C_{\bullet}$ and $D_{\bullet}$ be chain complexes of $R$-modules for a ring $R$, and let $M$ be an $R$-module.
It's known how to compute $\mathrm{Ext}_{R}^{*}(C_{\bullet},M)$; namely, there is a convergent Grothendieck spectral sequence whose second page is \begin{align*} E_{2}^{pq}=\mathrm{Ext}_{R}^{p}(h_{q}(C_{\bullet}),M)\Rightarrow\mathrm{Ext}_{R}^{p+q}(C_{\bullet},M). \end{align*} See, for instance, (13.8.2) in Remark 38 in these notes.
My question is what can be said when we replace $M$ with another complex $D_{\bullet}$.
Elaborating based on the comments: I am already familiar with spectral sequences that compute Tor; supposing $C_{\bullet}$ and $D_{\bullet}$ are bounded below: \begin{align*} ^{II}E_{pq}^{2}&&=&&\bigoplus_{q'+q''=q}\mathrm{Tor}^R_{p}(h_{q'}(C_{\bullet}),h_{q''}(D_{\bullet}))&&\Rightarrow&&\mathrm{Tor}_{p+q}^{R}(C_{\bullet},D_{\bullet})\\ ^{I}E_{pq}^{2}&&=&&h_{p}\mathrm{Tot}^{\oplus}\mathrm{Tor}_{q}(C_{\bullet},D_{\bullet})&&\Rightarrow&&\mathrm{Tor}_{p+q}^{R}(C_{\bullet},D_{\bullet}). \end{align*}
This comes from (5.7.5) in Weibel's An introduction to homological algebra. I would like to know the analogous statement for Ext, which Weibel skips.
In the particular case I care about, $C_{\bullet}$ is bounded below by $0$ (i.e., $C_{i}=0$ for $i<0$) and $D_{\bullet}$ is bounded -- specifically, $D_{j}=0$ for $j\not\in\{0,1\}$. Using a spectral sequence to compute explicitly even the first two or three Ext groups in this case would be fantastic.
If the general argument is well-known, please point me to it; otherwise for now I am happy to learn how to compute $\mathrm{Ext}_{R}^{n}(C_{\bullet},D_{\bullet})$ in terms of the homologies $h_{i}(C_{\bullet})$ and $h_{j}(D_{\bullet})$, even just for $n=0,1,2$.
EDIT: My suspicion/expectation is that the two spectral sequences are \begin{align*} ^{II}E^{pq}_{2}&&=&&\bigoplus_{q'+q''=q}\mathrm{Ext}_R^{p}(h_{q'}(C_{\bullet}),h_{q''}(D_{\bullet}))&&\Rightarrow&&\mathrm{Ext}^{p+q}_{R}(C_{\bullet},D_{\bullet})\\ ^{I}E^{pq}_{2}&&=&&h_{p}\mathrm{Tot}^{\prod}\mathrm{Ext}^{q}(C_{\bullet},D_{\bullet})&&\Rightarrow&&\mathrm{Ext}^{p+q}_{R}(C_{\bullet},D_{\bullet}), \end{align*} but (1) I do not know if this is correct, and (2) I am having a bit of trouble tracking the low-index convergence to calculate $\mathrm{Ext}^{0}_{R}(C_{\bullet},D_{\bullet})$, $\mathrm{Ext}^{1}_{R}(C_{\bullet},D_{\bullet})$, and $\mathrm{Ext}^{2}_{R}(C_{\bullet},D_{\bullet})$ for the aforementioned bounded below $C_{\bullet}$ and bounded $D_{\bullet}$.