The concordance function (from copula theory) can be defined as follows:
$$ Q = 4 \int \int_{\mathbf{I}^2} C_2(u,v) dC_1(u,v) -1 $$
My question is regarding how to evaluate the integral, and mainly the $dC_1(u,v)$ portion. I think this is a functional derivative, but I am unsure how to evaluate it for any given function $C_1$ (as I don't know functional analysis). Is this equivalent to the copula density $c_1(u,v) = \frac{\partial^2}{\partial u \partial v} C_1(u,v) $, or is it something else?
It has nothing to do with a functional derivative.
The notation $dC(u,v)$ implies integration with respect to the so-called C-measure where the measure of a rectangle $R_{ij} =[u_{i-1},u_i]\times [v_{j-1},v_j]$ is given by
$$V_C(R_{ij}) = C(u_{i},v_{j})- C(u_{i-1},v_{j})- C(u_{i},v_{j-1})+ C(u_{i-1},v_{j-1}) .$$
If the copula is absolutely continuous then the integral reduces to
$$\int \int_{\mathbf{I}^2} C_2(u,v) \, dC_1(u,v) = \int \int_{\mathbf{I}^2} C_2(u,v) \, \frac{\partial^2C_1(u,v)}{\partial u \partial v}\,du \,dv.$$