You will want to start by going into your Account Settings under the tab "Sharing & Privacy"

Next to where it says "Facebook" you will want to click "Add Account". You will then click the blue "Connect with Facebook button"

A popup window should request special permissions for the Raptr Connection. 

After logging in, you will receive 2 windows. One will ask to publish to your Facebook stream, the other will ask about updating Raptr regarding your contact email address on Facebook. 

You must approve publishing to your stream, or Raptr will not be able to post to your profile.

(Special Note: Once you click the 'connect with Facebook' button, all of the resulting screens you will see are coming from Facebook, not Raptr. One of the most common problems with these screens is that they are not sized correctly and you sometimes will not be able to see the buttons you need to click to approve special permissions. To correct this, simply click anywhere inside the window and hit the Tab key on your keyboard 4 times. This should scroll you to where these buttons are. Closing these windows without approving permissions will block Raptr from being able to post to your Facebook Profile.)