Let $K(S)$ be the category of chain complexes of $S$-modules; this category has Hom-complexes $hom_S(-,-)$ making it a dg-category, and thus produces a quasicategory via the construction in Higher Algebra section 1.3.1.
Take $f: S \to R$ a ring homomorphism, $M \in K(S)$, and $N \in K(R)$. Then there is a natural adjunction isomorphism between hom-complexes $$ hom_S(M, f^* N) \cong hom_R(R \otimes_S M, N) $$ which therefore induces an adjunction $K(S) \rightleftarrows K(R)$, and a monad $f_* f^*$ on $K(S)$ whose underlying functor is $R \otimes_S (-)$. This gives us a lift of the functor $f^*$ to the category $Alg(f_* f^*)$ of algebras for this monad (HA, 4.7.3.3).
For any object $M$ and $N$ in $K(R)$, we thus get a simplicial resolution $$ M \leftarrow R \otimes_S M \leftleftarrows \dots $$ The resulting complex $$ Hom_R(R \otimes_S M,N) \to Hom_R(R \otimes_ R \otimes_S M, N) \to \dots $$ can therefore be interpreted as calculating $Hom_{Alg(f^* f_*)}(f^* M, f^* N)$, because that's how one calculates homotopy classes of maps in the category of algebras over a monad. In the case where $M$ and $N$ are discrete complexes concentrated in fixed degrees, this recovers the relative Ext-groups you were interested in.
This adjunction between $K(S)$ and $K(R)$ is very unlikely to be monadic except in special circumstances. But there is a localizing subcategory $V \subset K(S)$ consisting of those complexes whose images under $R \otimes_S (-)$ are trivial, and in total a factorization of $f^*$ into maps of stable $\infty$-categories as follows: $$ K(S) \to K(S) / V \to Alg(f^* f_*) \to K(R) $$ Here $K(S) / V$ is the Bousfield localization / Verdier quotient / nullification of $V$. This is roughly a coimage / image factorization and one could ask whether the middle map is an equivalence; this I don't know.
Whether there are useful spectral sequences I also do not know (quotient maps $S \to S / I$ indicate that there is a cap on how much nice behavior we should expect). We can certainly construct filtrations of $R$ as an object of $K(S)$ and try to run with that, but it is typically much easier to reduce calculations in the derived category $D(S)$ to calculations in $K(S)$ and not the other way around.