An easy way to remember how to approach a character analysis is by using the acronym CID: Comprehend, Interpret, and Draw Conclusions.
Comprehension is gaining a basic understanding of what you are reading. Some questions to ask yourself may include, "Do I know what the plot is? Do I know who the characters are and how they interrelate and interact? Do I know where the story takes place and in what historical, social, and cultural setting(s)?"
Interpretation takes a deeper look at the details of the story. This is where simple knowledge begins to turn to understanding. Some questions to ask in this phase may include, "How does this character's self-perception clash with the way the author or narrator describes him/her? How does the setting contribute to the protagonist's/antagonist's choices? How do the characters' choices reflect their social positions? What purpose does each character serve in the story?"
As you attempt to answer these kinds of questions, you begin to draw conclusions about the characters, which then leads to your ability to write an analysis. It is as simple as CID!