**Edit**: Modified in accordance with Tom Leinster's entirely reasonable objections.

Sorry to exhume this question from 5+ years ago. In case someone is still looking for an answer, note that a very specific version (restricting to $V = \text{Cat}$) is addressed up to homotopy in the paper

M Bullejos and A Cegarra, *On the geometry of 2-categories and their classifying spaces*. K-theory, 29:211 – 229, (2003).

using the Duskin/Street nerve construction. Given a $V$-enriched category $C$, one constructs the simplicial set $\Delta C$ as follows: vertices are the objects of $C$, and higher simplices spanning objects $x_0,\cdots, x_d$ consist of

- 1-morphisms $f_{ij}:x_i \to x_j$ for $0 \leq i \leq j \leq d$, and
- 2-morphisms $\alpha_{ijk}:f_{ik} \Rightarrow f_{jk} \circ f_{ij}$ for $0 \leq i \leq j \leq k \leq d$
subject to certain associativity and identity constructions (see the Introduction of the linked paper for details).

One can also construct the Segal nerve as outlined in Chris Schommer-Pries's answer, but it is not as directly related to the objects and morphisms of the underlying enriched category. Bullejos and Cegarra show in the linked paper that the two constructions are naturally homotopy-equivalent so it doesn't matter too much either way!