Here is the full list of all 16 Murder... achievements.
Unlock policeman difficulty
Unlock a new theme
Fail a case
Solve your first case
Solve 25 cases on the highest unlocked difficulty
Solve 100 cases on the highest unlocked difficulty
Unlock private investigator
Fail 10 cases
Fail 50 cases
Solve a case in less than a day as detective sergeant
Solve a case in under two days as a private investigator
Solve 50 cases on the highest unlocked difficulty
Unlock detective sergeant
Solve a case in less than a day as detective inspector
Solve a case in under two days as detective chief inspector
Unlock detective chief inspector