I have played a few detectives in GURPS and my current favorite character is in a Monster Hunters type supernatural horror campaign that is in hiatus (the GM likes to skip around between several campaigns, so we will get back to it eventually). Running a good detective takes a lot so here are some ideas.
GURPS Resources
GURPS Mysteries is the go to book for playing a detective, especially a procedural one. It covers a lot of material, real world and fictional mystery solving, useful traits, GM tips, etc.
GURPS Action 2: Exploits is good for a cinematic and more fast paced investigation.
GURPS Classic: Cops is from GURPS Third Edition so some of the gear may need rework but goes into a lot of detail on law enforcement and playing someone from those professions.
GURPS Social Engineering is very helpful at the social aspect and this is an important part of most investigations.
Transhuman Space: Transhuman Mysteries is a good resource for science fiction mysteries, even if you do not use that setting.
For the GM
Before allowing a player to build a detective you should make a few decisions on how you want to run the campaign and let the player know.
Will Mystery be Important?
If solving mysteries are important to the plot you need to make sure the players have a good chance of solving them. So make sure they have key traits that will be useful and not too many things that will get in the way or derail the plot. Ideally you want the players to work at it, too easy and they are unsatisfied and too hard and they get frustrated. This will be easier to manage in a group where you know the players.
If mysteries are not important, let he player know so they do not get their hopes up. Managing Expectations (above) is a helpful resource here.
Time Management
If everyone is participating in solving the mystery that is terrific. However in most groups you will have your puzzle solvers and your combat and action types. The time needs to be managed so no single player is left sitting out most of the game session, much less the majority of them!
If only one player is trying to find clues and solve the mystery you need to keep the time short, or do individual sessions. An easy way to do this is to handle things with die rolls, such as skills or even the use of spells and powers. This is likely to be unsatisfactory to the player hoping to figure things out though. So set proper expectations.
What Traits?
What traits are allowed depend on the genre and setting. However, as GM you have a lot of wiggle room. Some abilities seem to be supernatural, yet with the right modifiers could be merely cinematic. Oracle (Digital), Psychometry and senses with the Hype sensory modifier are good examples of taking an obviously supernatural trait and making it "merely" cinematic. Sherlock Holmes instead of Lord Darcy for example.
High skill levels and especially certain powers or spells can make many investigations trivial. They also make the character feel special. In a campaign where the mystery is just a quick plot device and not meant to be something to spend considerable time on researching supernatural and cinematic abilities are the GMs friend. They let you easily justify the character finding the next link or whatever you need them to do and keep things moving.
In a more mundane setting they may not be allowed and if you want the players to take a lot of time investigating this is a good thing. But even here something like Intuition can help nudge things along as needed. Especially if one player is the detective and the rest are waiting around for the fighting.
For the Player
So you want to be a detective? GURPS really lets you shine in this roll and in a lot of possible ways.
You really need to get with the GM and be sure your on the same page as to what rules will be used and how much of the campaign will be focused on finding things out. If you want to be a detective and the GM plans on simple mysteries or wants them solved in a few minutes time you should put fewer points into these abilities. Even if much of the time is solving things you likely will want other skills so keep that in mind! Also if others are going to be involved what will they be doing while you investigate?
The first decision is what style and type of detective you want to be. GURPS Mysteries lists common archetypes and goes into skills and advantages that can be useful. Note that Spiritual Advisor costs the same as a 150% Ally and should probably be built as such. GURPS Action 2: Exploits is more for fast paced gaming but offers solid advice that is useful at any pace. Both supplements will be useful in building your character.
Since I am not aiming to write Mysteries 2: The Detective here as a blog post I wont go over much more but want to stress this again. It is very important to be on the same page as the GM and to not take too much screen time away from others.
How to share?
Hand tasks to offscreen Contacts or Patrons. Many things can take a lot of time and be boring to play out, Contacts can do a lot of that for you. Also you do not need high reliability levels for most tasks.
Get others involved. That stealthy ninja type? Let them break into things to get clues. The mage or psychic? They can find out information normal methods can't (though countermeasures are likely for known powers). The muscleman or deadly warrior can be your backup in risky meetings. Fighter types often have high HT and can be good at Carousing, let them find out basic info or even lure suspects into a trap for capture.
Sometimes You Have to Do it Yourself
If you never get to use your skills what was the point of paying for them? Again, be on the same page as the GM! A good GM will let you do some stuff on your own but do not expect it to happen all the time, nor for very long. When its your private moment try to be well prepared and have your modifiers and questions ready to go. Also, try to be entertaining for the others at the table. Most players will want thier own screen time but they will be a lot happier when not doing anything if the ones who are at least put on a good show. Its really boring to watch a bunch of die rolls and someone take minutes to decide what questions to ask or how they are going to break into something.
Prepare common mission items in advance. Its usually faster to tweak to the situation than to come up with a plan on the fly. Also have page references handy.