Let me sketch the definition of homotopy limit in full generality. Suppose $\mathscr{M}$ is a category with weak equivalences. Denote $\operatorname{Ho}(\mathscr{M})$ the category obtained by inverting weak equivalences. For any small category $I$, denote $\mathscr{M}^I$ the category of functors $I\rightarrow \mathscr{M}$. Defite weak equivalences in $\mathscr{M}^I$ to be the natural transformations between functors whose values are weak equivalences in $\mathscr{M}$. The 'constant diagram' functor $\mathscr{M}\rightarrow \mathscr{M}^I$ preserves weak equivalences, therefore it defines a functor, $$\text{constant diagram}\colon \operatorname{Ho}(\mathscr{M})\longrightarrow \operatorname{Ho}(\mathscr{M}^I).$$ The homotopy limit functor, $$\operatorname{holim}_{i\in I}\colon \operatorname{Ho}(\mathscr{M}^I)\longrightarrow \operatorname{Ho}(\mathscr{M}),$$ if it exists, is the right adjoint of the previous functor.
Notice that homotopy limits depend on the weak equivalences we consider. You have mentioned one of the weak equivalences you can take on DG-categories. There are other very interesting weak equivalences that you could also consider, and that would yield different homotopy limits.
The model category techniques show the existence of homotopy limits under certain hypotheses and tell us how to construct them from resolutions. A good reference is:
MR1944041 (2003j:18018) Hirschhorn, Philip S. Model categories and their localizations. Mathematical Surveys and Monographs, 99. American Mathematical Society, Providence, RI, 2003. xvi+457 pp. ISBN: 0-8218-3279-4 (Reviewer: David A. Blanc), 18G55 (55P60 55U35)
Suppose now that $\mathscr{M}$ is the model category of DG-categories you consider, and let $\mathscr{N}$ be the model category of $k$-linear categories where weak equivalences are $k$-linear equivaleces of categories. We can regard any $k$-linear category as a DG-category concentrated in degree $0$. The inclusion $\mathscr{N}\subset\mathscr{M}$ is a left Quillen functor with right adjoint
$$H^0\colon\mathscr{M}\longrightarrow\mathscr{N}.$$
This and the uniqueness of adjoints can be used to show that
$$H^0(\operatorname{holim}_{{i\in I}}\mathscr{C}_i) = \operatorname{holim}_I H^0(\mathscr{C}_i)$$
Here the second homotopy limit is in $\mathscr{N}$ which is hopefully easier to compute since the model category structure on ${\mathscr{N}}$ is simpler. I guess that further simplifications depend on particular cases you may want to consider.