Let $(\mathcal{C},W)$ be a pair with $\mathcal{C}$ a category and $W$ a wide (containing all objects) subcategory. Such a pair represents an $(\infty,1)$-category. One model for such gadgets is a quasi-category (a simplicial set satisfying the weak Kan extension condition). What is the direct procedure that constructs such a quasi-category from $(\mathcal{C},W)$? (I don't mind assuming that $(\mathcal{C},W)$ is part of a model structure if it simplifies things.)
I can do it indirectly. For example, given a model category, one can use the Dwyer-Kan technology to construct a simplicial category (by simplicial localization, hammock localization or whatever), apply fibrant replacement in the bergenr model structure for simplicial categories (i.e. making the mapping complexes Kan) and then take the simplicial nerve (Lurie, HTT 1.1.5). Another way is to construct a complete Segal space by Rezk's nerve construction and then take the zero row (note that this involves a fibrant replacement in the Reedy model structure). Both methods are quite complicated and I would like to know a more explicit construction. In particular, I would like to understand what are, say, the 0-, 1- and 2-simplixes of the resulting quasi-category.