I am just posting my comment as an answer. I will assume that we are working over a specified field $k$. I will also assume that $C$ is reduced, irreducible, normal, and proper. In the general case, you can first ask your question for the normalizations of the irreducible components of a proper model, and then you can try to descend from that normal, proper scheme to your original scheme.
The fiber product $Z_0 =C_0\times_{D_0} C_0$, considered as a closed subscheme of $C_0\times_{\text{Spec}\ k} C$, is a well-formed zero cycle of some degree $d$. There is an induced morphism $\zeta_{Z_0}:C_0\to \text{Sym}^d_k(C)$. There is an extension of $C_0\to D_0$ to a finite morphism with domain equal to $C$ if and only if $\zeta_{Z_0}$ extends to all of $C$. In turn, this holds if and only if the closure $Z$ of $Z_0$ in $C\times_{\text{Spec}\ k} C$ is finite with respect to the projection $\text{pr}_1:Z\to C$.
The articles of David Rydh are a great source for Chow varieties.