How to construct a quasi-category from a category with weak equivalences?

Every step in the following procedure is explicit, if somewhat complicated:

1. Construct the hammock localisation $L^H (\mathcal{C}, \mathcal{W})$. (See [Dwyer and Kan, Calculating simplicial localizations] for details.)
2. Apply $\mathrm{Ex}^\infty$ to every hom-space of $L^H (\mathcal{C}, \mathcal{W})$; this yields a fibrant simplicially enriched category $\widehat{L^H} (\mathcal{C}, \mathcal{W})$ because $\mathrm{Ex}^\infty$ preserves finite products, and the natural weak homotopy equivalence $\mathrm{id} \Rightarrow \mathrm{Ex}^\infty$ yields a Dwyer–Kan equivalence $L^H (\mathcal{C}, \mathcal{W}) \to \widehat{L^H} (\mathcal{C}, \mathcal{W})$. (See [Kan, On c.s.s. complexes] for details.)
3. Take the homotopy-coherent nerve of $\widehat{L^H} (\mathcal{C}, \mathcal{W})$ to get a quasicategory $\hat{N} (\mathcal{C}, \mathcal{W})$. (See [Cordier and Porter, Vogt's theorem on categories of homotopy coherent diagrams] for details.)

Let me make a few remarks to get you started.

• The objects in $L^H (\mathcal{C}, \mathcal{W})$ are the same as the objects in $\mathcal{C}$, and the morphisms are "reduced" zigzags of morphisms in $\mathcal{C}$.
• The natural weak homotopy equivalence $X \to \mathrm{Ex}^\infty (X)$ is bijective on vertices, so the Dwyer–Kan equivalence $L^H (\mathcal{C}, \mathcal{W}) \to \widehat{L^H} (\mathcal{C}, \mathcal{W})$ is actually an isomorphism of the underlying ordinary categories.
• The vertices (resp. edges) of $\hat{N} (\mathcal{C}, \mathcal{W})$ are the objects (resp. morphisms) in $\widehat{L^H} (\mathcal{C}, \mathcal{W})$, which are the same as the objects (resp. morphisms) in $L^H (\mathcal{C}, \mathcal{W})$.

The 2-simplices of $\hat{N} (\mathcal{C}, \mathcal{W})$ are harder to describe. Conceptually, they are homotopy-coherent commutative triangles in $\widehat{L^H} (\mathcal{C}, \mathcal{W})$, so they involve a simplicial homotopy in $\widehat{L^H} (\mathcal{C}, \mathcal{W})$; and by thinking about the explicit description of $\mathrm{Ex}^\infty$, the simplicial homotopies in $\widehat{L^H} (\mathcal{C}, \mathcal{W})$ are essentially zigzags of simplicial homotopies in $L^H (\mathcal{C}, \mathcal{W})$, i.e. zigzags of "reduced hammocks of width 1".