Adeel answer is perfect, I will be more basic.
There is many notions of "2-category" structure here around.
1) the category of small dg-categories $\mathbf{dgCat}$ is symmetric monoidal closed category. Closed means that there is an internal $HOM$ in the sense that for any two small dg-categories $A$ and $B$ there is $HOM(A,B)\in \mathbf{dgCat} $ such that there is a natural isomorphism of sets $\mathbf{dgCat}(X,HOM(A,B))\cong \mathbf{dgCat}(X\otimes A,B)$. That means $\mathbf{dgCat}$ is enriched over it self.

There is a functor from $H^{0}:\mathbf{dgCat}\rightarrow \mathbf{Cat}$ which gives you an enrichment of the category $\mathbf{dgCat}$ over $\mathbf{Cat}$, hence you can see $\mathbf{dgCat}$ as a 2-category.

On an other hand, the internal $HOM(-,-)$ described before has the wrong homotopy type, you can not derive it since it does not take Dwyer-Kan equivalences (between fibrant-cofibrant objects) to Dwyer-Kan equivalences (the $\mathbf{dgCat}$ is not symmetric monoidal model category in the sense of Hovey). Bertand Toen constructed, for the model category $\mathbf{dgCat}$, the right notion of the derived internal hom denoted by $RHOM(A,B)\in \mathbf{dgCat}$ (using bimodules, I will not write the details). Moreover this new derived $RHOM(A,B)$ induces the derived Mapping space $Map
_{\mathbf{dgCat}}(A,B)$ via the nerve functor of some well choosen subcategory of $RHOM(A,B)$). This new derived internal allows you to see the category $\mathbf{dgCat}$ as $(2,\infty)$-category and in the same time as symmetric monoidal $(1,\infty)$-category.

An important consequence is the following isomorphism in $Ho(\mathbf{sSet})$:
$$Map_{\mathbf{dgCat}}(A\otimes^{L}B,C)\cong Map_{\mathbf{dgCat}}(A,RHOM(B,C))$$

strict2-category, and hence has an underlying category. $\endgroup$2more comments