I think I pinpointed the problem. It is not system related but I think it has something to do with Adobe Flash player.
I too have this problem. When I first started the game, everything froze. I couldn't move my mouse. I thought my GPU bugged so I tried again with the same result. This got me thinking so I tried it third time but this time I ran Task Manager and put it to performance tab to see CPU and memory usage. When the client showed up and I logged in it froze again and Task Manager also was stuck which pinpointed it to being a bug in the client.
Fourth time I first ran some other games to see if something is not wrong with my system just to be sure. Every other game runs well which means it is League only. So I turned League of Legends again but this time with AIMP playing some music. To my surprise music kept playing and everything else froze. That means there is some memory problem, as if something is hogging all the RAM. Just like when you run something really heavy and everything freezes up for a short time.
I just kept waiting and soon computer unfroze and I got an error message in league client itself that says there was some server issue and they sent a report to tech team bla, bla, bla...
As I read posts from this thread I found that some people claim it was a problem with OS update, but I had no update in weeks. Everything is the same as it used to be, the only thing I updated was Adobe Flash player. Since the client runs on that retarded Adobe air and when I logged in every time, all the videos and animations in client were black. As if there was some problem with displaying content.
I will test it a little bit more but I think that Adobe Flash player did something so that Adobe air client bugs.