Define your character. Is she a timid, mousy young lady? Or is he a bully? Have a definite idea of the personality traits the character will have. This is important, because the narrative arc your character will follow will inform how he changes during his journey. By the end of the story, your character will be the exact opposite of what he was at the start. The timid, mousy young lady will become a self-assured, sexy woman. The bully will become more compassionate and understanding.
Plot your story. The plot will determine the narrative arc of your character. How will she change by the end of the story? This will be answered in the plot. For instance, in the movie "Casablanca," Rick Blaine is a cynical and self-interested club owner during the height of World War II. Ilsa, his onetime love, turns to him for help. Her husband is a member of the resistance, and both she and her husband are desperate to get out of Casablanca before the Nazis discover him. The only way to do that is to get fake passports. Rick, who knows the local riffraff, can do that for her, but his cynicism (due largely to Ilsa's betrayal of him years ago) and refusal to get involved in world events prevent him from initially helping Ilsa and her husband. But he eventually realizes there are more things at stake than his own hurt feelings, so he agrees to help them in the end. When the movie ends, Rick has changed from a cynical and self-interested man to someone who is more socially conscious and involved. Rick's narrative arc comes about because of the plot (his helping Ilsa and her husband get out of Casablanca).
As you plot your story, force your character to act in ways that are contrary to his initial nature. Let's say he's a cowardly young man who suddenly finds himself in the world of intrigue when he is mistaken for a dead spy. When other spies try to kill him, he tries to retrieve his former identity and get out of trouble, but is ultimately thrust into a plot to undermine a mastermind villain. The plot (defeating the villain) forces the protagonist to act courageously. When a spy tries to kill him, he is forced to defend himself. By the end of the story, the young man will have responded bravely in enough situations that he will no longer be a coward, but a courageous and self-assured man.
Make the transition your character undertakes believable. Show that the character isn't that far removed from the person she becomes at the end of the story. In "Casablanca," we are shown flashbacks of Rick's romance with Ilsa in Paris, which reveal who he was before he became cynical. His transition becomes believable because the person he becomes at the end of the movie isn't different from who he used to be. Consequently, the coward can show glimpses of bravery before the plot kicks in, demonstrating to the audience that he already had it in him, but simply needed to be put in the right circumstances to bring out his courageousness.
Show your character's psychological response to how she is changing. Rick, for example, initially rejects Ilsa's pleas for help. This creates the inner turmoil and tension the character will be feeling as the story progresses. Or, the character might actually enjoy this change. The coward, for instance, is no longer the poor schlub everyone ignored, and is now a very important and desirable man. As you can see, the character's narrative arc is dependent on the plot, but it's important that you show the character's transition in order for it to make an impact in your story.