$\def\E{\mathbf{E}}\def\V{\mathbf{Var}}\def\T{\mathbf{Tr}}$This may not be particularly helpful, but what you have is called simply the Generalized chi-square distribution.
You can calculate some of its moments without too much difficulty (you should check the calculations, though...)
$\E(x'Cx) = \E(C_{ij}x_ix_j) = C_{ij}\delta_{ij} = \T(C)$
$\begin{align} \V(x'Cx) & = \E(C_{ij}C_{mn}x_ix_jx_mx_n) - \E(x'Cx)^2\\ & = C_{ij}C_{mn}\E(x_ix_jx_mx_n) - \T(C)^2 \\ & = C_{ij}C_{mn} (\delta_{ij}\delta_{mn} + \delta_{im}\delta_{jn} + \delta_{in}\delta_{jm}) - \T(C)^2 \\ & = \T(C\,'C) + \T(C^2) \end{align}$
If you substitute $C=I$ then you recover $\E(x'x)=n$ and $\V(x'x)=2n$, as you'd expected for a regular chi-square distribution.
The Wikipedia article implies that the distribution is equivalent to a sum of independent chi-square distributed random variables, but this is not obvious to me from the definition.