In practice, one tends to "compute" arbitrary homotopy colimits as bar constructions, especially when you have a simplicial model category.
If $X:J\to P$ is a simplicially enriched functor, where $J$ is small, then you get a "bar construction" $B=B(*,J,X)$. This is a simplicial object in $P$, with
$$B_0 = \coprod_{j_0\in \mathrm{ob}J} X(j_0),$$
$$B_1=\coprod_{j_0,j_1\in \mathrm{ob}J} X(j_0)\times J(j_0,j_1),$$
$$B_2=\coprod_{j_0,j_1,j_2\in \mathrm{ob}J} X(j_0)\times J(j_0,j_1)\times J(j_1,j_2),$$
etc.
(Here "$\times$" really means the simplicial "$\otimes$"; if $P$ is simplicial sets, then it really is $\times$.) If $X$ is suitably cofibrant, then the realization $|B|$ of $B$ will be the homotopy colimit of $X$.
This bar construction I described above is really a special case of "use the projective model structure"; you can use a bar construction to an explicit construction of a projective cofibrant resolution of $X$ (typically under some hypothesis on $X$, such as that each $X(j)$ is cofibrant in $P$). In fact,
$$|B(*,J,X)| = \mathrm{colim}_J |j\mapsto B(J(-,j),J,X)|,$$
and there is a weak equivalence $|B(J,J,X)|\to X$, which is a true projective cofibrant approximation given some mild hypothesis on $X$.
The standard references are oldies but goodies: Segal's paper "Classifying spaces and spectral sequences," IHES 1968, and the "yellow monster": Bousfield & Kan, "Homotopy Limits, Completions, and Localizations," LNM 304.
Added:
When $J=\Delta^{\mathrm{op}}$, you can say something easier: the homotopy colimit of $X: J\to P$ is computed by the realization $|X| \in P$ (again, up to the cofibrancy of the objects $X(j)$). I don't know an explicit reference offhand, though everybody uses this fact; it may be in the two that I cited.
Best Answer
Let $\mathscr C$ be a small category. Necessary and sufficient conditions for a presheaf $F$ to be cofibrant in the global projective model structure on $[\mathscr C^\mathrm{op}, [\Delta^\mathrm{op}, \mathbf{Set}]]$ are that:
(1) Each $F(-)(n) \colon \mathscr C^\mathrm{op} \to \mathbf{Set}$ is projective (i.e., a coproduct of retracts of representables; if $\mathscr C$ is Cauchy-complete, then equivalently a coproduct of representables).
(2) $F \colon \mathscr C^\mathrm{op} \to [\Delta^\mathrm{op}, \mathbf{Set}]$ factors through the subcategory $[\Delta^\mathrm{op}, \mathbf{Set}]_{\mathrm{nd}}$ of $[\Delta^\mathrm{op}, \mathbf{Set}]$ whose objects are simplicial sets and whose maps are simplicial maps which sends non-degenerate simplices to non-degenerate simplices.
On the one hand, it's easy to show by induction that any cofibrant object satisfies (1) and (2). Conversely, condition (2) means that each $F(-)(n)$ breaks up as a coproduct $G_n(-) + H_n(-)$ of non-degenerate and degenerate simplices, and the object $G_n(-)$ of non-degenerate simplices is projective since $F(-)(n)$ is. This means that any object satisfying (1) and (2) can be built up dimension by dimension using the generating cofibrations: first construct the $0$-skeleton using pushouts of maps
$\partial \Delta_0 \cdot \mathscr C(-, W) \to \Delta_0 \cdot \mathscr C(-, W)$ (for $W \in \mathscr C$)
one for each representable summand in the projective object $F(-)(0)$; then construct the $1$-skeleton by using pushouts of maps
$\partial \Delta_1 \cdot \mathscr C(-, W) \to \Delta_1 \cdot \mathscr C(-, W)$ (for $W \in \mathscr C$)
one for each representable summand in the projective object $G_1(-)$ of non-degenerate $1$-simplices; and so on.