= Solution
Apply <Green's second identity> to $\Phi(\mathbf r)$ and $G(\mathbf r,\mathbf r')$, differentiating in $\mathbf r$:
$$
\int_V(\Phi\Delta G-G\Delta\Phi)\,dV
=\int_S\left(\Phi\frac{\partial G}{\partial n}-G\frac{\partial\Phi}{\partial n}\right)dS.
$$
The volume terms become $\Phi(\mathbf r')-\int_VG\rho\,dV$. On the boundary $G=0$ and $\Phi=f$, so
$$
\boxed{\Phi(\mathbf r')=\int_V\rho(\mathbf r)G(\mathbf r,\mathbf r')\,dV
+\int_S f(\mathbf r)\frac{\partial G}{\partial n}(\mathbf r,\mathbf r')\,dS}.
$$
Here $\partial/\partial n$ is the outward <normal derivative> in the integration variable. One may justify the identity at the singularity by excising a small sphere around $\mathbf r'$; its unit flux supplies the evaluation $\Phi(\mathbf r')$.
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