Character Portraits are when you are interviewing someone and you are able to push X to zoom in on their features and scan them for details. You'll know when the Portrait is complete when all the blurred writing in the top right corner is uncovered and it returns to the interview. The portrait icon at the bottom of the screen should turn green when completed.
To get this, you need to get every Character Portrait in the game without missing any. The furthest right entry in the casebook shows how many portraits you have unlocked, as well as how many there are for the case you are currently on.
If you end the Final Case too early, by fingering the wrong suspect, you will miss out on meeting the last character required for this achievement. You must progress to the Circus to meet this character.