The main purpose of a character study or character analysis essay is to shine some insight on that characters significance, purpose, or nature. This means that it should go beyond simply the biographical inforation of the character. That information is, however, a reasonable place to start your analysis. The character's name, age, social status, and/or location may all be briefly described in order to begin painting the picture of who it is you will be studying in your essay. It is also common to include what type of character you are dealing with. Is it a protagonist or an antagonist? Is your character static or dynamic? How do you know? The bulk of your paper, though, should be dedicated to information that goes beyond just these facts that demonstrate your basic understanding of who the character is.
Character studies require a focus on characterization, or the processes an author uses to convey information about characters in the story. There are two basic types of characterization: direct and indirect. Direct characterization is the use of descriptions that directly relate to the character. For example, if somewhere in a story an author writes, "Sally had a generous nature and loved to help all who were in need," that author is using direct characterization to show the reader that Sally is generous. Indirect characterization is when an author shows rather than tells us a trait. Let's say that in this same example story the author did not just write that Sally was generous. Instead, the author writes about how Sally delivered homemade quilts to the homeless shelter, and on her way there she decided to adopt a stray kitten, and after she left the homeless shelter she gave a pair of her own shoes to an orphan she found in an alley. The author never says Sally is generous, but the reader is given the sense that Sally is generous based on Sally's own actions. That is an example of indirect characterization. Your paper should explore both types of characterization for your particular character, but the traits you choose should relate to a particular point you are trying to make about your character. Why is your character important to the story? What part does (s)he have in the conflict, climax, and/or resolution? A character study is not just about who a character is, but what a character's place is within a story. Any characterization you discuss in your essay should help the essay's readers to understand the characters significance.
If you are looking for some more ideas as to what character traits might be significant when writing your essay, you should take a look at the eNotes page on Character Analysis that I have linked below.
These are the general elements to focus on in a character study essay If you need any help with the elements specific to your own essay, I would be happy to look it over for you and help in that respect. Please send me a message if that is the case.