Hi. I got into beta yesterday so I installed for the first time last night. I'm not 100% sure the best way to report things but I wanted to write this stuff down while it was fresh in my head before forgetting it. I'm just going to list out a few issues I've had.
Issue 1. I chose custom installation and chose the Directory it should install to as "C:\Games\League of Legends". The final install path that LoL ended up in was C:\Games\League of Legends\Riot Games\League of Legends". It's a minor issue but would be nice if instead of just prepending my install root it would replace the "Riot Games\League of Legends" portion of the default install path.
Ok, after that LoL patched up fine, I played 3 games and logged off for the night. Tonight I went to play by launching from the icon installed to my desktop.
Issue 2. Every time I tried to launch I got a pop up error dialog that said something along the lines of "ASSERT:Pointer null" and then had a path in it like "..\..\..\..\src\somethingsomething". Unfortunately I didn't screenshot this. To fix it I ran the launcher to uninstall and reinstall the game fresh. I chose the default install location this time and it seemed to start up fine after patching, just like last night. I was able to create a game and play... I was also able to close the game completely and relaunch it from the desktop.
I'm playing under Windows 7. Not sure what other info would be useful to you, hope this post helps. I may uninstall and reinstall in a custom location again tomorrow to see if I can recreate.