I got the same problem: When I am downloading the game from PlayOnLinux, I am getting a message to not click on "Launch League of Legends". When I try to start it from the icon on my desktop(to start patching the game) first try I get a Bug Splat, second and other times when I try after first I got a window that I can't clearly see it, something about "Another Instance of League of Legends is already running would you like to close it?" and when I try to close it, it gives me an another window.... My linux version is: Linux Mint 17.3 Cinnamon 32-bit, Rosa I think..
Did you follow our guide when installing League of Legends? http://www.gamersonlinux.com/forum/threads/league-of-legends-guide.206/