I will answer the title question about computing homotopy limits. Let $A : \mathcal{C} \to \mathbf{Cat}$ be a small (strict) diagram. By expanding the definitions of the cobar construction, one eventually discovers that $\operatorname{holim} A$ can be computed as the end
$$\int_{[n] : \mathbf{\Delta}} \left[ \mathbf{I} [n], \prod_{(c_0, \ldots, c_n)} \mathcal{C} (c_n, c_{n-1}) \times \cdots \times \mathcal{C} (c_1, c_0), A c_0 \right]$$
where $\mathbf{I} [n]$ is the contractible groupoid with $n + 1$ objects. In more familiar terms, this is just the hom-category of all morphisms between a certain pair of cosimplicial categories.

Here is a description of an object in $\operatorname{holim} A$:

- For each object $c$ in $\mathcal{C}$, we have an object $a_c$ in $A c$.
- For each morphism $f : c_1 \to c_0$ in $\mathcal{C}$, we have an isomorphism $\mu_f : f (a_{c_1}) \to a_{c_0}$ in $A c_0$; and $\mu_{\mathrm{id}_c} = \mathrm{id}_{a_c}$ for all objects $c$ in $\mathcal{C}$.
- For each composable pair $f_1 : c_2 \to c_1, f_0 : c_1 \to c_0$ in $\mathcal{C}$, we have a commutative triangle in $A c_0$,
$$\begin{array}{ccc}
f_0 (f_1 (a_{c_2})) & \rightarrow & f_0 (a_{c_1}) \\
& \searrow & \downarrow \\
&& a_{c_0}
\end{array}$$
i.e. $\mu_{f_0 \circ f_1} = \mu_{f_0} \circ f_0 (\mu_{f_1})$.
- For each composable triple in $\mathcal{C}$, we have a commutative tetrahedron in $A c_0$.
- etc.

In fact, the coherence conditions above degree 2 are automatic if all the triangles commute. Thus we see that an object in $\operatorname{holim} A$ is the same thing as a pseudocone over $A$ (where we regard $A$ as a pseudofunctor with the canonical coherence data). Unsurprisingly, the morphisms are the same, so $\operatorname{holim} A$ (as constructed above) is the pseudolimit of $A$.

subgroupoidof $[\mathcal{A}, \mathcal{B}]$ as the simplicial hom-space $\underline{\mathrm{Hom}}(\mathcal{A}, \mathcal{B})$: see this note of Rezk. Thus one can use the standard technique of bar and cobar constructions: see e.g. this preprint of Shulman). Note that the existence of injective/projective model structures is not needed. $\endgroup$ – Zhen Lin Feb 19 '14 at 10:25Sketches of an elephant, Part B] shows that a certain non-full subcategory of $\mathbf{\Delta}$ is 2-coinitial. $\endgroup$ – Zhen Lin Feb 19 '14 at 21:30