Study the material in which the character appears. Read the entire book, or watch the entire movie, before going back to study the character for your essay. This provides a good background for noticing any nuances or characteristics that you will want to write about in your essay.
Write the introduction to your character essay. Introduce the character to the reader, including the name. Depending upon the level of writing, you may write about his appearance, his history, and any habits or attitudes that were instrumental into the storyline.
Most character essays involve a writing prompt. An example of this would be "Why did Susan run away?" You can use what you know of the character to outline why Susan ran away instead of facing the obstacle.
Organize your subsequent paragraphs so that they describe the character in a logical manner. Every paragraph should be focused on the character and the writing prompt. This will show your reader that you understand the character and the issues you are studying.
Conclude your character essay by using facts. Statements should be backed up by facts, not just opinions. This will show your reader that you understood the character and his reactions to the given situation.