I think you should look into the theory of fundamental solutions, or Greens functions. Greens functions are solutions to the equation

\begin{equation}
G_{xx}+G_{yy}=\delta
\end{equation}

Which are explicitly know in this case (I think it is $\frac{\log(x^2+y^2)}{4\pi}$, but you have to look this up). Solutions of the original equation are then

\begin{equation}
U=G\star f+U'
\end{equation}

With $\star$ the convolution and $U'$ a solution to the homogeneous Laplace equation
\begin{equation}
U_{xx}'+U_{yy}'=0.
\end{equation} Of course this requires some regularity of $f$. If $f$ is a compactly supported distribution this has a distributional solution. If one assumes a certain smoothness as well, this will induce smoothness of the solution.

More on this theory can be found in the chapter on fundamental solutions of

MR2680692 Duistermaat, J. J. ; Kolk, J. A. C. Distributions.
Theory and applications.
Translated from the Dutch by J. P. van Braam Houckgeest.
Cornerstones. Birkhäuser Boston, Inc., Boston, MA, 2010. xvi+445 pp. ISBN: 978-0-8176-4672-1