Small dg-categories are categories enriched over $\mathcal{V} = \textbf{Ch}(k)$, the category of chain complexes over a commutative ring $k$.

It is a general result that if $\mathcal{V}$ is (co)complete, then so is the category $\mathcal{V}\textbf{-Cat}$ of small $\mathcal{V}$-enriched categories. See the text around Corollaries 7.3.6 and 7.3.7 of Christina Vasilakopoulou's thesis for a sketch of the proof and other references of this fact.

The way this paper by Harvey Wolff proves the cocomplete case is by first considering $\mathcal{V}$-graphs, which are just $\mathcal{V}$-categories without identities or composition. The construction for coproducts is easy enough, but things get quite involved when constructing coequalizers.

In the remainder of this answer, I'll just show how to construct coproducts of $\mathcal{V}$-categories.

Let $\mathcal{C}_i, i \in I$ be a collection of $\mathcal{V}$-categories for some indexing set $I$. Form a $\mathcal{V}$-category $\mathcal{C}$ in the following manner:

• $\text{Ob}(\mathcal{C}) := \coprod_{i \in I} \text{Ob}(\mathcal{C}_i)$

Small dg-categories are categories enriched over $\mathcal{V} = \textbf{Ch}(k)$, the category of chain complexes over a commutative ring $k$.

It is a general result that if $\mathcal{V}$ is (co)complete, then so is the category $\mathcal{V}\textbf{-Cat}$ of small $\mathcal{V}$-enriched categories. See the text around Corollaries 7.3.6 and 7.3.7 of Christina Vasilakopoulou's thesis for a sketch of the proof and other references of this fact.

This paper by Harvey Wolff proves that $\mathcal{V}\textbf{-Cat}$ is cocomplete by first showing that $\mathcal{V}\textbf{-Graph}$ is cocomplete ($\mathcal{V}$-graphs are just $\mathcal{V}$-categories without identities or composition). The construction for coproducts is easy enough, but things get quite involved when constructing coequalizers even for just $\mathcal{V}$-graphs.

In the remainder of this answer, I'll just show how to construct coproducts of $\mathcal{V}$-categories.

Let $\mathcal{C}_i, i \in I$ be a collection of $\mathcal{V}$-categories for some indexing set $I$. Form a $\mathcal{V}$-category $\mathcal{C}$ in the following manner:

• $\text{Ob}(\mathcal{C}) := \coprod_{i \in I} \text{Ob}(\mathcal{C}_i)$
• For $x,y \in \text{Ob}(\mathcal{C})$ where $x \in \text{Ob}(\mathcal{C}_i)$ and $y \in \text{Ob}(\mathcal{C}_j)$, define $\mathcal{C}(x,y)$ to be $\mathcal{C}_i(x,y)$ if $i = j$ and $0$ otherwise, where $0$ is the initial object of $\mathcal{V}$.
• For $x,y,z \in \text{Ob}(\mathcal{C})$, we need to produce composition maps $$\mu_{x,y,z}\colon \mathcal{C}(x,y) \otimes \mathcal{C}(y,z) \to \mathcal{C}(x,z).$$ If $x,y,z$ are all from the same $\mathcal{C}_i$, then just use the composition maps from $\mathcal{C}_i$. Otherwise, at least one of $\mathcal{C}(x,y)$ and $\mathcal{y,z}$ is $0$. Since $0 \otimes X = X \otimes 0 = 0$ for any $X \in \mathcal{V}$, the domain of $\mu_{x,y,z}$ is $0$, so take $\mu_{x,y,z}$ to be the unique map $0 \to \mathcal{C}(x,z)$.

It's easy to check that composition as defined above is associative and unital.