I am essentially just repeating Will's answer, but giving a slightly different point-of-view (and a relevant reference).

Let $F_r$ be a free group of rank $r>1$.  Given a discrete faithful representation $\rho:F_r\to \mathrm{PSL}(2,\mathbb{R})$ it is torsion-free and so $\mathbb{H}^2/\Gamma$ is a complete hyperbolic surface $\Sigma_\Gamma$, where $\Gamma:=\rho(F_r)$.  You asked how to determine the homoeomorphism type of $\Sigma_\Gamma$.

In general, you need to know the gluing data for the fundamental domain.  For this, it suffices to know simple closed curves around each boundary and puncture (which tells you $n+b$).  For then, using $r=2g+n+b-1$ you can determine $g$.

Around each collar neighborhood of a boundary, the holonomy $\rho$ must correspond to a translation (which determines a geodesic length) and so its absolute trace will be $>2$.  Around each puncture the holonomy of the corresponding loop must be a horolation (rotation at infinity) and so its absolute trace must be 2.  So from the traces you can determine which loops are at punctures and which are at boundaries.

I recommend reading concrete examples in *[Trace Coordinates on Fricke spaces of some simple hyperbolic surfaces][1]* by William M. Goldman.  See Section 4, in particular.


  [1]: https://arxiv.org/abs/0901.1404