Go to the directory of game, \Batman Arkham City\Binaries\Win32.
Right-click on BATMANAC and go to Properties. Click on the Compatibility tab and check Run this program as an administrator. Then, delete xlive.dll from the folder.
Now, go to C:\Program Files\Microsoft Games for Windows - LIVE\Client or the appropriate path for your computer, and right click on GFWLIVE. Make sure that program runs as an administrator as well.
Finally, start Batman and press the Home button.