Determine first if the story needs to be told in the first person. Frequently, a writer will discover during the writing process that her story is really a third person point of view story and not first. Ask yourself before writing whether your character has a unique voice that will serve your narrative aims. Will your story be enhanced if it is told only through your character's point of view? Does your character have a unique perspective that will enliven the narrative and plot?
Establish your character's voice. What does she sound like? What I mean by this is: does your character have a particular way of expressing herself? Is she sarcastic, for instance? Does she have a sense of humor? Or is she pompous? Is she articulate or inarticulate. Is she a teenager, elderly? Or is she an immigrant struggling to get a handle on a language not native to her own? All of these things must be taken into consideration when establishing your character's voice. Take a look at Vladimir Nabokov's Lolita or Alice Walker's The Color Purple. The characters in those novels have a very precise way of speaking that adds to the narrative thrust of the stories.
Maintain your character's voice. Look out for slips in the narration. Choose your language carefully. If your character is inarticulate or uneducated, then he probably won't use elevated or Latinate language. On the other hand, if your character is a pompous intellectual, then she most likely won't speak the latest street slang. Unless change within a character is a part of your story (such as Charly in The Flowers of Algernon), try to maintain your character's narrative voice throughout.
Don't slip into omniscience. Omniscient voice simply means that your character, like God, sees and knows all, including what other characters are thinking. This is not a first person point of view, which is limited only to what the narrator thinks and feels. Of course, a first-person narratorcan speculate what other characters are thinking and feeling, but this should only be used to reveal something about your protagonist and not necessarily about other characters in the story. For instance, your narrator could imagine what her boyfriend is thinking while they are checking out apartments, revealing the character's insecurities or expectations in her relationship.
Make your character interesting and lively. There's nothing worse than a boring First Person narrator. The character doesn't have to be bizarre or eccentric to be interesting. Just give him or her a unique perspective on life or a unique and interesting way of expressing himself. A good example of this is J.D. Salinger's Holden Caulfield in The Catcher in the Rye.