You've set the stage for the mystery. The victim has met his demise and your detective is here on the scene ready to fix the crime. But is he ready to complete his job?
You have created your detective, given him a personality and an appearance. You've plotted an offense for him to solve and given him a team to work with. Now you need to provide him the appropriate mental tools to aid in his quest.
Certain mental abilities are necessary to handle any situation and it is particularly so for your detective. Your detective doesn't need to be perfect, but to ensure that him to have been promoted to detective; he needed to own these skills. That does not mean he still has them, you are able to withhold some of the follow to accommodate your story's plot.
I believe this is simply not a whole list, but it will give you a concept of what it takes mentally to be always a detective, which will help you write your detect better.
Detectives can be found in many personalities and appearances, but each of them share some basic mental skills that make them do their job.
Observation: Detective must have a highly developed ability to see what others miss. He needs to have the ability to look at the crime scene or clues and piece together what happened.
Writing Ability: As your detective's reports will undoubtedly be utilized in court, he should have the ability to write, and speak, concisely and clear.
Desire: Your detective should have an all natural curiosity and need to resolve the crime and catch the bad guy. Not because it's his job, but because his heart tells him.
Confidence: Whether it is speaking with witnesses or arresting the bad guy, your detective should not hesitate or show fear.
Self-Directed: A detective needs to have the ability to work without close supervision and make decisions on the move. Much of his work is done on the fly, minus the luxury of getting a manager nearby.
Patience: Detectives deal with a lot of various things, from working with victims/witnesses to stakeouts and long court proceedings.
Open-Minded: Your detective should let the important points of the crime, not their personal, religious or moral beliefs speak.
Intelligence: A detective needs to have above-average intelligence. His memory must be excellent. He should be able to think logically and with a little imagination when focusing on a crime.
As I said earlier, you possibly can make your detective anyway you wish, but they're the traits most police departments look for when promoting an officer to the rank of detective.