Research private eyes in your setting. There are different laws about private investigators in different states and countries --- ranging from whether or not they need a license to what it takes to get a license and to what they need for an investigation that crosses state lines.
Use familiar traits and tropes wisely and avoid clichés. Tropes are familiar elements that can help to set the mood for your story --- like the overcoat and fedora worn by private eyes in noir fiction. While they can help you communicate with the reader, if you use too many tropes, the reader will feel as if he has seen it all before.
Describe your character --- include everything from general information like height, weight and ethnicity to more specific details like birthmarks, tattoos and manner of dress. This in-depth and precise description of your character will help you to communicate effectively these character traits to the reader.
Write your character's history and decide what made him decide to be a private investigator. The more effort you put into a character's history, the more fodder you have to determine his thoughts and reactions during the course of a story. People draw their opinions, views and perceptions from their own experiences, so your private eye needs to have had experiences.
Figure out your character's speech patterns. Does he like to explain random trivia? Does he make a lot of references to music, books, politics or history? Is your character more likely to pontificate or does he keep conversation to a minimum and play things close to the vest?
Explore your character's behavior and temperament with a character interview. It may sound strange to "interview" a fictional character, but in answering questions as your character, you learn his voice. You get a better idea of how he feels about things, what motivates him, and what he wants out of a life. All of that will transfer to your story.