Here is the full list of all 16 Murder... achievements.
Solve your first case
Fail a case
Solve 25 cases on the highest unlocked difficulty
Solve 100 cases on the highest unlocked difficulty
Unlock a new theme
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 policeman 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