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wine: Call from 0x7b83eb22 to unimplemented function api-ms-win-crt-runtime-l1-1-0.dll._initialize_onexit_table, aborting
err:module:attach_process_dlls "MSVCP140.dll" failed to initialize, aborting
err:module:LdrInitializeThunk Main exe initialization for L"C:\\Riot Games\\League of Legends\\RADS\\projects\\league_client\\releases\\0.0.0.50\\deploy\\LeagueClient.exe" failed, status 80000100

I heard there was a way to override the 

MSVCP140.dll file somehow, but I haven't found an actual fix. Have checked LoL's official forums, and the LoL PoL subforum, no answers in either location. 

Have you checked this out? : https://www.reddit.com/r/leagueoflinux/comments/5tboyt/uptodate_guide_for_the_new_client_using/

I personally haven't tried it yet, but hopefully it can get you up and running :)