Is dgCat a category or a 2-category? Let us consider dgCat, the "collection" of all small dg-categories. In On differential graded categories and Lectures on dg categories the authors state that they form a category, i.e.  dgCat has small dg categories as objects and the dg functors as morphisms. They then give a model structure on dgCat.
However, as pointed in What do DG-categories form?, it is more likely that dgCat is a 2-category: it does not make too much sense to say whether two dg-functors $F$ and $G$ are equal.  Instead we can talk about morphisms between dg-functors. 
How to integrate these two viewpoint? In particular, could we still have a model category structure on dgCat if we treat it as a 2-category?
 A: 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 $Ho(\mathbf{sSet})$:
$$Map_{\mathbf{dgCat}}(A\otimes^{L}B,C)\cong  Map_{\mathbf{dgCat}}(A,RHOM(B,C))$$
A: The model structure on the category of dg-categories presents an $(\infty,1)$-category DGCat.  This structure is essentially provided by the existence of mapping spaces (or mapping $\infty$-groupoids) between dg-categories.
To see why DGCat admits the further structure of an $(\infty,2)$-category, it is sufficient to see why these mapping $\infty$-groupoids can be refined to mapping $(\infty,1)$-categories, which give back our mapping $\infty$-groupoids when we pass to the sub-$\infty$-groupoids of invertible morphisms.
Such structure is provided by functor dg-categories, which are the internal hom objects in DGCat (in the "derived" sense -- these are denoted $R\underline{Hom}$ by Toen).  These functor dg-categories give $(\infty,1)$-categories (by using for example the Dold-Kan correspondence), which provide the mapping $(\infty,1)$-categories in the $(\infty,2)$-category DGCat.
(Toen showed that these mapping $(\infty,1)$-categories can also be described as the $(\infty,1)$-categories of right quasi-representable bimodules.)
Edit: I just noticed that the preprint [Giovanni Faonte, $A_\infty$-functors and homotopy theory of dg-categories, arXiv:1412.1255] appears to contain a precise construction of the $(\infty,2)$-category of dg-categories.
