I am sorry to hear you are having this problem. The first thing I recommend is verifying the integrity of the game cache, to see if this is a file based issue or a network connection issue.
Go to C:\Program Files(x86)\Steam\SteamApps\Common\Call of Duty: Modern Warfare 3\Players2, and delete the singleplayer and multiplayer cfg files.
Once these files are deleted,
Go to the Steam Application, and select Library
Right click on Call of Duty: Modern Warfare 3 and select Properties > Local Files tab > Verify Integrity of Game Cache
Relaunch the game and see if this resolves the problem. If you are still having trouble beyond this, then we may need to do some network troubleshooting.