Let $A_*$ and $B_*$ be simplicial algebras over a simplicial commutative ring $R_*$. I would like to understand how one explicitly computes the derived tensor product $A_* \otimes^L_{R_*} B_*$. More specifically, I am interested in the case where $R_*$, $A_*$, and $B_*$ are constant simplicial rings $R$, $A$ and $B$, in which case it appears to be well-known that $\pi_n(A_* \otimes^L_{R_*} B_*) = \operatorname{Tor}^R_n(A,B)$.
I am aware that the general method is to apply a suitable cofibrant replacement, but I do not understand how one computes them. I am moreover aware of the spectral sequence outlined in Theorem II.6.6 of Quillen's Homotopical Algebra, which is strong enough to show what I am after, but I am entirely unable to follow it.
In the case that $R_{\bullet}$, $A_{\bullet}$, and $B_{\bullet}$ are constant simplicial commutative rings, one can use the bar construction. Fix a monad $T:\mathcal{C}\to\mathcal{C}$ for which the $T$-algebras are $R$-algebras. For an $R$-algebra $S$, the bar construction $B(T,S)_{n}$ is $T^{n+1}S$ with face/degeneracy maps induced by the action of $T$.
The bar construction, when $T$ is a "free" functor, creates a simplicial resolution $B(T,S)_{\bullet}\to S$, hence a cofibrant replacement in the model category structure which can be used to compute $S\otimes^{L}T_{\bullet}:=S^{c}\otimes T_{\bullet}$, where $S$ is a constant simplicial algebra and $S^{c}$ is any cofibrant replacement of $S$.
In general, the bar construction is rather unwieldy (the standard is the polynomial resolution where $T(S)=R[S]$). That shouldn't be too surprising though, for the same reason constructing projective resolutions of modules is unwieldy in the general case, even with finiteness tools like Hilbert's Syzygy Theorem. Fortunately, there are some pet examples where things are manageable, like $T:R\mbox{-}\mathbf{alg}\to R\mbox{-}\mathbf{alg}$, $T(S)=S\otimes_{R}R[y]$, where the $T$-algebras are $R[y]$-algebras.
See http://math.uchicago.edu/~amathew/SCR.pdf, section 4, for some bar constructions and a great example computing $\pi_{n}(R\otimes_{R[y]}^{L}R)$ using the manageable bar construction mentioned in the paragraph above. See https://arxiv.org/pdf/math/0609151.pdf, section 4, for the same exercise in a slightly different flavor, referenced by Mathew's paper.