The maximum rank throughout LA Noire is 20 and shouldn't be missed if you're using a guiding walk-through.
Throughout your cases, you'll gain XP Points consisting of 4 requirements which includes:
1. Completing your Cases (Unsure of XP points Earned) But shouldn't be missed since story related.
2. Branch Correct Questions = 5XP
3. Completing Street Crimes = 15XP
4. Finding Landmarks and Film Reels = 5XP
You should be able to earn this trophy when you reach the end of L.A. Noire Story.
If you don't branch correct questions, try again to contribute other trophies or do street crime when you have the time and find a few landmarks.