I'd like to have some simple examples of quasi-categories to understand better some concepts and one of the most basic (for me) should be the category of chain complexes.

Has anyone ever written down (more or less explicitly) what the simplicial set corresponding to the quasi-category associated with the category of (say unbounded) chain complexes on an abelian category looks like?

I am not looking for an enhancement of the derived category or anything like this, I'm thinking of the much simpler infinity category where higher morphisms correspond to homotopies between complexes. My understanding is that the derived category should then be constructed as a localization of this $\infty$-category.

I am guessing my problem lies with the coherent nerve for simplicial categories.