There is an interpretation of homology using derived categories, and you can find a brief treatment around remark 3.3.10 in Kashiwara-Schapira "Sheaves on Manifolds". If $M$ is an $n$-manifold and $a: M \to \ast$ is the tautological map to a point, then the homology of $M$ with coefficients in an abelian group $F$ is the homology of the complex $Ra_! a^! F$. If we write $or_M = H^{-n}(a^!F)$ for the orientation sheaf, then the counit of the adjunction induces an "integration" map $Ra_! a^! F \to F$ that induces a map $H^n_c(M,or_M) \to F$. When $F$ is a real vector space and $M$ is smooth, you can relate this to differential forms, because the orientation sheaf is quasi-isomorphic to a suitable version of the de Rham complex. Softness of the de Rham sheaves implies you get an isomorphism $H^n_c(M, or_M) \to \frac{\Gamma_c(M, \Omega^n \otimes or_M)}{d\Gamma_c(M,\Omega^{n-1} \otimes or_M)}$. Stokes's theorem on $M$ implies a comparison between the "integration" map and "honest integration" of forms (or densities), in particular, the fact that $d\Gamma_c(M,\Omega^{n-1} \otimes or_M)$ is annihilated by honest integration.

I don't think Stokes's theorem is a statement whose proof is made easier with derived categories. Instead, I would rather say that it implies the fact that honest integration actually yields a morphism in a derived category.

Here's the setup: If we are given a smooth manifold $M$ and a $k$-form $\omega$, we can obtain a real number from any smooth map $f$ from the standard $k$-simplex $\Delta$ to $M$, by integrating $f^\ast\omega$ along $\Delta$. This produces a map of graded real vector spaces $\int_M: \Omega^\ast(M) \to \operatorname{Hom}(\operatorname{Sing}_\ast(M),\mathbf{R})$.

Here's Stokes's theorem: $\int_M$ is in fact a map of cochain complexes.

If you want to prove the theorem efficiently, you can use naturality of pullback to reduce to a simpler statement about forms on $\Delta$ itself. There will always be a step where you have to use properties of integrals. It is common to invoke Fubini's theorem (e.g., the short proof in Guillemin-Pollack), but you can do a lower-tech proof using the fact that polynomial forms uniformly approximate smooth forms on the simplex.

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