Content on this page is probably outdated and represents my personal knowledge, feelings and understading of things at that time.
Getting League Of Legends working on a 64bit Archlinux installation with ati open source drivers (xf86-video-ati) is actually quite straightforward once you know what to install (presuming you already have multilib system) :).
First, you need to have your open source ati graphic drivers set up correctly. This is out of scope, but well documented on Archlinux Wiki[1].
Second, you need to install some 32bit libraries:
# glxinfo32 | grep direct
# glxinfo32 | grep GL_EXT_texture_compression_s3tc
Third, open Play On Linux, find League of Legends in Install (note: check the Testing checkbox) and follow the on-screen instructions.