\begin{equation*}
-\Delta u+u=(I_{\alpha}\ast|u|^{p})|u|^{p-2}u+\lambda|u|^{q-2}u
\end{equation*}
where $N\geq 3,\ \alpha\in(0,N)$, $\lambda>0$, $q\in(2,\frac{2N}{N-2}]$, $p=\frac{N+\alpha}{N}$ or $\frac{N+\alpha}{N-2}$ are the critical exponents in the sense of Hardy-Littlewood-Sobolev inequality and $I_\alpha$ is the Riesz potential. Based on the results of the subcritical problems, and by using the subcritical approximation and the Poho\v{z}aev constraint method, we obtain a positive and radially nonincreasing groundstate solution in $H^1(\mathbb{R}^N)$ for the problem. To the end, the regularity and the Poho\v{z}aev identity of solutions to a general Choquard equation are obtained.