Hey Y'all,
Just Installed Windows 10 and LoL didn't work, this is how I fixed it.
Go ahead to your LoL directory. In my case [ F:\Riot Games\League of Legends ] and run the "lol.launcher.admin".
If you are still getting the error try going to Windows Firewall [ Settings -> Network & Internet -> Window Firewall ], Now scroll through the list to see if you have any LoL files being allowed. In my case I found nothing so had to manually add it.
Click "Allow an app or feature through Windows Firewall", click "Change Settings", click "Allow another app...", click "Browse...", navigate to your LoL directory and add "lol.launcher.admin" (you can allow both, I did).
Ok now try to run the "lol.launcher.admin".
IF it STILL doesn't work I have one more trick, try going back to your LoL directory. In my case [ F:\Riot Games\League of Legends\RADS\system ] and search for ''system.cfg'',
Open it with notepad++ (or just notepad should work too), delete everything on it (if there's anything) and paste this content:
DownloadPath = /releases/live
DownloadURL = l3cdn.riotgames.com
Region = EUNE
!!! IF YOU DON'T PLAY ON EU NORDIC & EAST SERVERS THEN THE LAST LINE WILL DIFFER FOR YOU !!!
Save it, and try running the launcher again.
GoodLuck!
P.S. I found that if the game is running in Compatibility Mode ( Mine was set to use Window 7 ), after clicking "Launch" in the Client screen it will lead you to a Black Login Screen. Just Disable it to fix :P.