In order to get this achievement you need the following conclusions:
- Stolen valuable papers
- Looking for papers
- Neligan's innocence
- Harmless flirting
- Unexplained presence
- Hurtley's innocence
- Feat of strength
- Pouch proves guilt
You then will be able to choose Cairns as the murderer. To be ale to get this achievement you have to choose the "send him to jail" option (or the show no mercy option as it's called in my language).
Cairns will come to Baker Street and start a fight - this is where it gets a little tricky. You have to aim at his face with the left stick and quickly press "A" when the circle turns blue. After three consecutive successful hits you have to press "Y" in a QTE to perform the final punch.
He will go down and your achievement should pop!
If you fail simply press "X" after the conversation with Lestrade to make use of the "Replay Ending" option.