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awesome, thats perfect...
hmmm not sure at this point.
Thanks for sticking with it!

Did you install Internet Explorer with Service Pack 3?
Don't forget to set Windows version: Windows XP
or it won't install.
I'm going to keep researching...

This is the log error when I run the game and it just start updating again and gets stuck (it doesn't download)

[08/06/13 19:16:00] - Running wine-1.5.24-LeagueOfLegendsShop lol.launcher.exe (Working directory : /home/pablo/.PlayOnLinux/wineprefix/leagueoflegends/drive_c/Riot Games/League of Legends)
fixme:ntdll:find_reg_tz_info Can't find matching timezone information in the registry for bias 240, std (d/m/y): 28/04/2013, dlt (d/m/y): 8/09/2013
( ERROR)[19:16:00.538] RADS::Common::RegistryHelp::RegKeyValueString::RegKeyValueString: Failed to find registry value "ProgramW6432Dir" (2).
fixme:ntdll:NtConnectPort (0x5b0c1170,L"\\\\ThemeApiPort",0x33fadc,(nil),(nil),(nil),0x33faec,0x33fae8),stub!
fixme:ntdll:find_reg_tz_info Can't find matching timezone information in the registry for bias 240, std (d/m/y): 28/04/2013, dlt (d/m/y): 8/09/2013

Check this out, I found this Guide: League of Legends Ubuntu 12.04

How:
1. Install wine 1.5.3 amd64 (from repository ppa:ubuntu-wine/ppa)
2. In winetricks i chouse:
a.) "Select default wine prefix"\\"change settings"
check boxes: "win7", "videomemorysize=512" (equivalented by my videocards memory), "vsm=hardware", "psm=enabled"
b.) "Select default wine prefix"\\"Install a Windows dll or component"
check boxes: Adobe Air 2.7, all starting from d3dx* and direct*, devenum, dinput, dinput8, dirac, dmsynth, directx9, dsound, dxdiag, dxdiagn, gecko120, ie8, msls31, pdh, physx, quartz, vcrun2008, vcrun2010, winhttp, wininet (Not all of them can be installed from winetricks on 1.5.3 64bit version of wine because it seems there is 64bit windows core and 32bit packages from older wine releases. If you have errors, try install all of them again except errored packages)".
c.) "Select default wine prefix"\\"Run regedit" i add REG_SZ key with name: "ProgramW6432Dir" and with value "C:\\Program Files" in "HKLM\\Software\\Microsoft\\Windows\\CurrentVersio n"
3. Additional i download this windows-packages and install it thrue wine: Adobe Air 3.2 (From Adobe.com site), Visual c++ redistributable 2005/2008/2010 (both x86 and x64 from download.microsoft.com totally 6 .exe files), Net. Framework 3.5 sp1 and Net. Framework 4.0 (x86/x64 from download.microsoft.com)
4. Just copy "League of Legends" directory from my WinXP machine to /home/user/.wine/drive_c/games/
5. In terminal: change current directory (with "cd" command) to /home/user/.wine/drive_c/games/"League of Legends"/RADS/system
6. Execute this command:
wine rads_user_kernel.exe run lol_launcher $(ls ../projects/lol_launcher/releases/) LoLLauncher.exe

LoL works fine with graphical settings on High except shadows (Medium) both in Borderless and Fullscreen mods.

He says use Wine 1.5.3 but obviously he's not using PlayOnLinux.
I noticed this part, related to your debug error:

c.) "Select default wine prefix"\\"Run regedit" i add REG_SZ key with name: "ProgramW6432Dir" and with value "C:\\Program Files" in "HKLM\\Software\\Microsoft\\Windows\\CurrentVersio n"

Its from over a year ago, but maybe the one Registry change will fix that error?

Did you install Internet Explorer 8?

Yes, everytime I installed League of Legends I installed IE8. I didn't get how to add "REG_SZ key with name: "ProgramW6432Dir" and with value "C:\\Program Files" in "HKLM\\Software\\Microsoft\\Windows\\CurrentVersio n" in winetricks

My install works without IE8, so it should work without it for you. However, the out of game store is just a black screen for me on the Wine1.5.24League version. When I use other versions of Wine, I get a certificate error at the out of store screen and sometimes a crash.

I had issues with the client crashing also, but I can't remember if it was the wininet or rasapi32 causing it. Make sure to use those also.

So is the game working now?
Yes other versions of wine will crash when going to the store in-game.
Wine 1.5.24-LeagueOfLegendsStore was the version I tested as a fix for the crashing and it worked.
But there are other League of Legends Wine versions too... try one of those.

I don't know for sure if League of Legends requires IE8, but I think it needs that Service Pack 3 and msxml3.
IE8 is just a quick way to get both at once.

Sometimes when I run the game I get the update screen (half the time it gets stucked and half the times it downloads the updates).

Rarely I get the loggin screen, but when putting my user name and passworn and clicking loggin, it freezes :(

I don't understand what it means "c.) "Select default wine prefix"\\"Run regedit" i add REG_SZ key with name: "ProgramW6432Dir" and with value "C:\\Program Files" in "HKLM\\Software\\Microsoft\\Windows\\CurrentVersio n""

Can you help me with that?

Sometimes when I run the game I get the update screen (half the time it gets stucked and half the times it downloads the updates).

Rarely I get the loggin screen, but when putting my user name and passworn and clicking loggin, it freezes :(

I don't understand what it means "c.) "Select default wine prefix"\\"Run regedit" i add REG_SZ key with name: "ProgramW6432Dir" and with value "C:\\Program Files" in "HKLM\\Software\\Microsoft\\Windows\\CurrentVersio n""

Can you help me with that?