There is an entire book dedicated to this theorem:Uniformisation des surfaces de Riemann-Henri Paul de Saint-Gervais= (Aurélien Alvarez, Christophe Bavard, François Béguin, Nicolas Bergeron, Maxime Bourrigan, Bertrand Deroin, Sorin Dumitrescu, Charles Frances, Étienne Ghys, Antonin Guilloux, Frank Loray, Patrick Popescu-Pampu, Pierre Py, Bruno Sévennec, Jean-Claude Sikorav.) http://www.amazon.fr/Uniformisation-surfaces-Riemann-th%C3%A9or%C3%A8me-centenaire/dp/2847882332/ref=sr_1_1?s=books&ie=UTF8&qid=1344144095&sr=1-1 A lot of historical insight!
Here you find some video-lectures on the same subject by one of the authors:Etienne Ghys:
http://video.impa.br/index.php?page=escola-de-altos-estudos-the-uniformization-theorem-old-and-new
