Writing a character analysis paper requires examining, usually the main character, or the protagonist in a detailed manner.
I would open the paper with a detailed summary of who the character is in the work, describing his role in the central plot. Then you should analyze the characters internal and external conflicts. Internal conflicts are character vs. self conflicts, external conflicts are character vs. character, or character vs. environment.
This part of the analysis brings into the discussion the character's relationships with other characters in the story. You can examine how the character changes throughout the story. You can make predictions on the character's future.
To end the story, I always tell my students to reflect on their opinion of the character. Did they like the character, do you know anyone like the character? Would you be friends with this person, why or why not?
Below is an excellent guide to writing a character analysis.
For a true character analysis, you don't need to waste your time summarizing that character's actions throughout the text. You can reference a couple of things and incorporate that information in your statements about that character in your intro. Note how my description of Edna Pontellier accomplishes this: "In Kate Chopin’s novel The Awakening, the consequences and scope of Edna Pontellier's arousing development reach beyond the limits of simple love into the richer depths of self-awareness and self-worth. Stifled by the implications of high society and bound to it by her marriage, Edna finds herself mindlessly moving through life’s paces until love’s spark finds its way into her heart for the first time with the sudden friendship of the young and handsome Robert Lebrun." This effectively conveys her character's journey while bequeathing contextual details.
Now, onto your analysis. Pick aspects of the character that overtly demonstrates his or her personality. Be sure that you have quotations from the text to back up these claims. Once you have established your opinion of this character in your intro, use the rest of your paper to strengthen your argument. Don't just say, "Edna Pontellier is a selfish protagonist" - explicate your statement by showing why and how she is this way. If her selfishness is the crux of your thesis, use your body paragraphs to highlight individual evidence of this behavior.
You begin by selecting the character you would like to write about. Once you have that decided then you make a list of what the character said, how the character felt, what they were thinking throughout the story, what they did in the story, and then what the problem was they encountered and their solution to the problem. This will help you start off to see if you have enough to write about. If you can't think of these things then...I suggest you choose another character and follow the same steps. I hope this helps you get started if you haven't done so already.