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.

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.

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.

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.