= Solution
Remove a small disc about $x$ and apply <Green second identity> to $u(x_0)$ and $G_0(x;x_0)$. Since $\nabla_{x_0}^2G_0=\delta(x_0-x)$ and $\nabla^2u=-\rho$, the shrinking inner boundary contributes $u(x)$ and gives
$$
u(x)=-\int_\Omega G_0(x;x_0)\rho(x_0)\,dx_0dy_0
+\oint_{\partial\Omega}\left(u\,\partial_nG_0-G_0\partial_nu\right)ds.
$$
For the unit disc, the <method of images> gives the <Dirichlet Green function>
$$
G_D(x;x_0)=\frac1{2\pi}\log\frac{|x-x_0|}{|x_0|\,|x-x_0/|x_0|^2|},
$$
which vanishes on the boundary by the hinted identity. The boundary representation with $u(1,\theta_0)=\delta(\theta_0-\alpha)$ evaluates the normal derivative of $G_D$ at $\theta_0=\alpha$, giving the <Poisson kernel>
$$
\boxed{u(r,\theta)=\frac1{2\pi}\frac{1-r^2}{1+r^2-2r\cos(\theta-\alpha)}.}
$$
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