Note [2018]:
Check the Online Fix & Community Mod Patch Guide to fix the black screen/crash problem and more!
I've tested their Community Mod Patch and it works without problems on Win 10.

I will keep this guide up if anyone needs it, because this solution still works:
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1: Install the game
( When I tried to start the game = black screen + error message.)

2: Uninstall the Games for Windows live files via -uninstall-
There are two entries, so uninstalled them both!

3: Restart PC. (not necessary?)

4: Download and install this version of GFWL (I didn't bother to fill in any email-adress etc, after the installation)
http://www.xbox.com/en-GB/live/PC/DownloadClient
http://download.gfwl.xboxlive.com/content/gfwl-public/redists/production/xliveredist.msi

(4a: I've got some suggestion about downloading the xlive.dll file and copy it into the directory of the game, but I haven't tried it myself)

5: Start the game.
Don't try to log into an existing account, since there will be no connection anyway.
If you try to save progress you might get a crash since it trys to connect to your GFWL account.
So when you start the game you get the GFWL window. Although I have a account. I just go to -> New profile -> Local profile (it's not a button, the link is in the text!).
Create a local account and use that to save your game.

Enjoy!
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