This is more a long comment rather than a complete answer.
I think existence/uniqueness depends so much on $V$, the domain $\Omega$ and the boundary datum $g:\partial \Omega \to \mathbb{R}$ that it's impossible to give a unified overview on if/when existence/uniqueness holds. I will outline a simple method that solves some cases and shows a bit the interaction between $V, \Omega$ and $g$.
I will assume $V \in L^\infty(\Omega)$ is smooth, $V>0$ and $g \in C^1(\partial \Omega)$. Let $\lambda_1(\Omega)>0$ be the first Dirichlet eigenvalue of the Laplacian in $\Omega$ and $w$ be the harmonic extension of $g$ in $\Omega$. We want to solve
$$ \cases{-\Delta u = V(u+w) \:\: in \: \Omega \,, \\ u=0 \: on \:\:\partial \Omega \,.} $$
If we can do so then $f:=u+w$ solves $-\Delta f = Vf $ as you want and has boundary datum $g$. Finding solutions is equivalent to finding critical points of
$$ J(u):= \int_{\Omega} |\nabla u|^2 - \int_{\Omega} V(u+w)^2 \,. $$
By the Poincaré inequality
$$ \int_{\Omega} V(u+w)^2 \le \frac{2\|V\|_{L^\infty}}{\lambda_1(\Omega)} \int_\Omega |\nabla u|^2 + 2\|V\|_{L^\infty} \int_\Omega w^2 \,, $$
hence
$$ J(u) \ge \left(1-\frac{2\|V\|_{L^\infty}}{\lambda_1(\Omega)} \right) \int_\Omega |\nabla u|^2 - 2\|V\|_{L^\infty} \int_\Omega w^2 \,.$$
This means that if $2\|V\|_{L^\infty} < \lambda_1(\Omega)$ then $J$ is coercive and one can find a solution by direct minimisation. Note that this is a condition on both $V$ and $\Omega$. To avoid trivialities let me assume $g$ is not identically zero so that $w\not \equiv 0$ and $\int_\Omega w^2 >0$. By the direct method you find there exists a minimiser $u_0$ of the problem, and this is not identically zero since $u_0 \equiv0$ does not solve our PDE. Then $u_0+w$ solves the problem you want.
Nevertheless, I have no good reasons to believe the minimiser is unique (let alone that there is only one critical point) even in this case, since the functional is not convex.
A further comment on existence. There are extremely effective ways of proving the existence of one (or many) solution coming from variational methods/min-max theorems, but every theorem is tailored on some specific types of problems and it's hard to give one that works in general. If you look on books they kind of look like the same but they are all a bit different. Every time you are in the position to say that a Palais-Smale sequence is bounded (one of such conditions is what I do above), then your specific functional satisfies the Palais-Smale condition and you can try some variational method.
If you have some more informations on $V$, $\Omega$ and the boundary datum $g$ I could try my best to see if I can show you the existence of one critical point.
