I think that I need to install the game without using Steam.
Not possible. Steam Community is integrated into the game and is required to run it. The packaging states "you must activate this product via the Internet by registering for a Steam account and accepting the SSA." SSA=Steam Subscriber Agreement. This includes allowing Steam to monitor what processes are running on your computer as an anti-cheating measure:
"Steam and the Software may include functionality designed to identify software or hardware processes or functionality that may give a player an unfair competitive advantage when playing multiplayer versions of any Software, other Valve products, or modifications thereof ("Cheats")."
The only thing I can really suggest without knowing anything about your computer is to try uninstalling and reinstalling Steam. A fresh install from a downloaded installer, not the one on the Portal 2 disc.
You can download a new installer here:
http://store.steampowered.com/about/?snr=1_4_4__11If you are still having problems, there's no one better than Steam's own support to help troubleshoot your issue. They can be contacted here:
https://support.steampowered.com/newticket.php