After Colbert bursts through your door at the start of the game and you've passed all the QTEs he will offer you a bribe (either take the bribe or not it doesn't matter), Go to a phone after he has left and call his wife to tell her that he isn't cheating.
Colbert will thank you and be a lot friendlier with later interactions meaning you're on the right track.
After you've snuck into O'Leary's Gambling rig and get caught by the red panda and letting O'Leary talk, use Colbert to your advantage by saying he asked you to come (which in turn will save his life and stop you from getting "The Reaper" achievement).
Later on when you're back in the office with Weekly, you'll get a phone call from Colbert's wife thanking you for saying he's loyal. Tell her to enjoy their marriage and you should get the achievement.