In https://kerodon.net/tag/00PK, Lurie introduces the dg-nerve functor from differential graded categories to simplicial sets as a tool to translate statements/ constructions from the dg-context to the context of $(\infty,1)$-categories, as it actually even takes values in those. But as is suggested in https://ncatlab.org/nlab/show/nerve+and+realization, most constructions of nerves and realizations tend to follow a general scheme, i.e. nerves usually have left adjoint realization functors that are defined as Kan extensions of, in this case, a functor I will call $DG: \Delta \rightarrow \operatorname{dg-Cat}$, so that the dg-nerve should be given in components by $N^{dg}_n(\mathcal{C})= \operatorname{Fun}(DG([n]),\mathcal{C})$, should this principle also apply to the nerve of dg-categories, as the name and the general philosophy would suggest.
However, I was neither able to find literature on some kind of dg-realization, nor was I able to construct a representing object $DG([n])$ compatible with the construction used by Lurie myself (I however suspect that this dg-category should have exactly n objects). That is a shame, as it would make a lot of things much easier. Has anybody read something about such a construction or maybe has an idea on how to do it, or does at least know a simple argument why there can't be one? Greetings from Heidelberg,
Intergalakti
To close this question off, I'll just say a few words about the approach in arxiv.org/pdf/1312.2127.pdf, the paper Kevin Carlson pointed out in the comments, so anyone stumbling upon this question gets a rough overview.
Indeed, there is a way to define the dg-nerve via such a construction, in fact there even are two. If one regards the formula that Lurie gives, it seems pretty difficult to give a construction that accounts for all of it, as there is next to the sum term also a composition and a boundary term. The reason for those seems to be that they are part of the infinite amount of higher compositions induced by the $A_\infty$-operad, and if one extends the category of dg-categories to the bigger realm of $A_\infty$-categories, there is a fairly natural construction of a generalization of the dg-nerve, going along the lines I described above.
Additionally, there is some kind of bigger dg-nerve, obtained by truncating the morphism complexes, applying Dold-Kan and then the homotopy-coherent nerve (this is also described in Luries "Higher Algebra"). Although this looks a mit more elegant at first glance, since one does not need $A_\infty$-algebras, the result is much migger, although equivalent to the first construction; I am a bit surprised that a truncation occurs as this way one loses some information, I thought the functor was fully faithful?
Anyway, especially the first approach is very helpful and explains the ad hoc formula that Lurie gives, from this point of view the theory of $A_\infty$-algebras seems like a very natural generalization of the dg-theory.