The black screen is the result of Riot poorly implementing that feature. Whoever designed it used Windows default windows which in a DirectX window such as League of Legends causes them to appear behind the black screen. This is known and why Riot hasn't fixed it is beside me.
Either way, if you have connection issues you'll want to check your firewall and all the other standard checks. Obviously your internet works, you posted this, but you may be blocked by your ISP or Firewall from connecting to Riot's Game Servers and that could cause this kind of issue. I solved this particular issue when I encountered it by opening my Services (Services under Control Panel or search Services on Windows 8). Scroll down to Windows Firewall and disable the service. If you use Windows 8 this will prevent you from accessing the Windows 8 store as this service handles some of the security features for that. To make sure it works open a custom game with yourself as the only player. If you can load you're set, if not close it and change something else and try again.