I've just taken a course which concluded with a sketch of the uniformisation theorem for Riemann surfaces, following the last chapter of Gamelin's Complex Analysis. The idea is:
-If the Green's function exists for your surface, use it to construct a conformal map from the surface to a bounded region in the complex plane. Now apply the Riemann mapping theorem.
-If the Green's function doesn't exist, construct a meromorphic variant called the bipole Green's function. Similar to the first case, we can use this to construct an injective map from the surface to the Riemann sphere. Now if this map misses more than one point of the Riemann sphere, simply-connectedness of the domain (and hence the image) means that the image is bounded, so we can use the Riemann mapping theorem. Otherwise the map misses one point (so the surface is conformally equivalent to the complex plane) or is surjective (conformally equivalent to the Riemann sphere).
Complex analysis is very far from my field, so I'm afraid I can't explain this any further (and I apologise for any inaccuracies in the above).