"Character Observations" as referred to by this achievement are when you press X while interviewing someone and the camera zooms in on their features and lets you move around their body to look out details about them.
You'll know when you're finished a character observation when all of the blurred lines are uncovered at the top right and it takes you back to the interview questions. At this point the small portrait icon at the bottom should be green to show you have finished that character.
To unlock this you should be observing EVERY character you interview as you progress thru the game. without failing to complete any of them. This also counts progress towards "The Methodical Man" achievement for collecting all character portraits.
The achievement should unlock during the third case "Blood Bath" as you talk to the three suspects in the case as long as you've been completing all portraits to this point.