I’d like to expand on the accepted answer and comments.

Cisinski describes in §7.5.25 of his book how to compute right derived functors in the context of an $\infty$-category $C$ with weak equivalences and fibrations. This is good enough to compute $RF$ if $F:K(A):=N_{\mathrm{dg}}(\operatorname{Ch}A)\to D$ with $A$ a Grothendieck abelian category and $D$ any $\infty$-category: one takes quasi-isomorphisms as the weak equivalences in $K(A)$ and the fibrations to be as in the model structure on $\operatorname{Ch}A$ of HA.1.3.5.3. That this satisfies the axioms of Cisinski’s Definition 7.4.12 is easy to check, since a square in $K(A)$ is cartesian iff it’s cocartesian iff the map on cones is an equivalence in $K(A)$; i.e. a chain homotopy equivalence (HA.1.2.4.14 & HA.1.3.2.17). If $D=D(B):=K(B)[\mathrm{q.i.}^{-1}]$ with $B$ an abelian category then this computes $RF$ for any additive functor $F:A\to B$.

What about tensor product? Well, in that case one would like to find cofibrations that together with the weak equivalences make $K(A)$ into an $\infty$-category with weak equivalences and cofibrations and so that the cofibrant objects are the K-flat complexes. I don’t know how to do this in general, but when $A=\operatorname{Mod}_R$ then a preprint of Gillespie shows there exists a model structure on $\operatorname{Ch}A$ whose cofibrant objects are the K-flat complexes. In any case, it seems at least superficially easier to compute with Deligne’s formula than to find the right fibrations or cofibrations, so let’s turn to that.

**Proposition**
Let $C$ and $W$ be as in §7.2.1 of Cisinski’s book and let $D$ be a (co)complete $\infty$-category. Suppose given a right (left) calculus of fractions $W(x)$ at $x\in C$ and let $\gamma:C\to C[W^{-1}]$ be the localization map. Then
$$(\gamma_*F)(\gamma x)\simeq\lim_{z_0\to x_0}F(z),$$
where the limit is indexed over $W(x)$
(respectively
$$(\gamma_!F)(\gamma x)\simeq\operatorname{colim}_{x_0\to z_0}F(z),$$
indexed over $W(x)$).

(Here, $\gamma_*:\operatorname{Fun}(C,D)\to\operatorname{Fun}(C[W^{-1}],D)$ is the functor of right Kan extension, so $\gamma_*F$ computes $LF$, while $\gamma_!:\operatorname{Fun}(C,D)\to\operatorname{Fun}(C[W^{-1}],D)$ is left Kan extension and $\gamma_!F=RF$.)

**Corollary**
Let $A$ be an abelian category, $D$ a cocomplete $\infty$-category, and $F:K(A)\to D$ a functor. If $K\in K(A)$,
$$RF(K)\simeq\operatorname{colim}_{K\to K’} F(K’),$$
where the (filtered) colimit is indexed by $K(A)_{K/}^{\mathrm{q.i.}}$, the full subcategory of $K(A)_{K/}$ on the quasi-isomorphisms. Dually, if $D$ is instead complete,
$$LF(K)\simeq\lim_{K’\to K}F(K’),$$
where the (cofiltered) limit is indexed by $K(A)_{/K}^{\mathrm{q.i.}}$.

The corollary follows from the proposition using Cisinski’s Theorem 7.2.16, as the set of quasi-isomorphisms is closed under composition as well as pullback and pushout in $K(A)$.

*Proof of Proposition –*
We can rewrite (the dual of) Cisinski’s Corollary 7.29 as follows: if $F$ is a functor $C\to\mathcal S(=\mathrm{Kan})$ and there is a right calculus of fractions $W(x)$ at $x\in C$, then
$$\lim_{z_0\to x_0}F(z)\simeq(\gamma_*F)(\gamma x),$$
where the limit is indexed over $W(x)$.
If $G:X\to Y$ is any functor of simplicial sets and $x\in X$, let $G_x:=x^*G$ denote the $x$-fiber of $G$. The functor
$$\gamma_*:=\operatorname{Fun}(X,\gamma_*):\operatorname{Fun}(X,\operatorname{Fun}(C,\mathcal S))\to\operatorname{Fun}(X,\operatorname{Fun}(C[W^{-1}],\mathcal S))$$
is functorial in $X$, so in particular $\gamma_*(\Phi)_x=\gamma_*\Phi_x$ for any $x\in X$ and functor $\Phi:X\to\operatorname{Fun}(C,\mathcal S)$. Letting $X=D^{\mathrm{op}}$ and $\Phi=h_DF=\operatorname{Map}_D(-,F):C\to\operatorname{Fun}(D^{\mathrm{op}},\mathcal S)$, we produce via right Kan extension a functor $\gamma_*(h_DF):C[W^{-1}]\to\operatorname{Fun}(D^{\mathrm{op}},\mathcal S)$ so that for each $d\in D$ and $x\in C$,
$$(\gamma_*(h_DF)_{\gamma x})_d=\gamma_*\operatorname{Map}_D(d,F)_{\gamma x}=\lim_{z_0\to x_0}\operatorname{Map}_D(d,Fz)=\operatorname{Map}_D(d,\lim_{z_0\to x_0}F(z)).$$
On the other hand, the functor
$$\alpha:\operatorname{Fun}(C[W^{-1}],h_D):\operatorname{Fun}(C[W^{-1}],D)\to\operatorname{Fun}(C[W^{-1}],\operatorname{Fun}(D^{\mathrm{op}},\mathcal S))$$
sends $\gamma_*F=LF$ to $\gamma_*(h_DF)$ by
Cisinski’s Proposition 6.4.9. Therefore
$$(\gamma_*(h_DF)_{\gamma x})_d=(\alpha(LF)_{\gamma x})_d=\operatorname{Map}_D(-,LF(\gamma x))_d=\operatorname{Map}_D(d,LF(\gamma x)).\qquad\square$$