When writing a character analysis, what we are looking for is exactly what the character is like as a person. There are several things that can be examined in order to complete a character analysis. Below is a link to a larger list of the things to examine, but some of them are the character's ethics, motives, whether or not we think the character's actions are wise or unwise, and whether or not the character's actions create any effects. Since we are limited to space, we won't be able to discuss all of these details for Elizabeth in Pride and Prejudice, but below are a few ideas to get you started.
For one thing, we do know that Elizabeth is a very ethical character with a strong sense of morality. In fact, we see her as a critic of not only herself but of her parents as well. Elizabeth criticizes herself when, after reading Darcy's letter, she finally realizes just how wrong she has been in her judgements. She judged Wickham to be admirable and trustworthy, simply because he is friendly and conversational. While she judged Darcy to be prideful, arrogant, and conceited well before she met Wickham, she also let her wrongful trust in Wickham influence her opinion of Darcy, thereby judging Darcy to be the most despicable person she's ever met. However, Darcy's account of Wickham's actions help her to see things rightly. She knows that Darcy's account is trustworthy because the story involves his own sister, and he certainly wouldn't slander his sister just to put Wickham in a negative light. As Elizabeth phrases her self-revelation:
How despicably have I acted! ... I, who have prided myself on my discernment!--I, who have valued myself on my abilities! who have often disdained the generous candour of my sister, and gratified my vanity, in useless or blameable distrust ... Had I been in love, I could not have been more wretchedly blind. (Ch. 36)
Hence, we see from her self-judgement that Elizabeth is characterized as having a strong sense of what is right and wrong and as valuing moral behavior. She now sees that her pride, her judgements, and her vanity have definitely been immoral behavior.
We can also analyze Elizabeth's character from any effects she has created. For example, her chastisements of Darcy when he first proposes, though misguided, were actually beneficial for him. Darcy was absolutely repulsed and mortified by the idea that she viewed him as arrogant, conceited, selfish, and that his proposal was delivered in a less "gentleman-like manner" than what would be expected of him (Ch. 34). As a result, Darcy not only explains his behavior, but also makes every effort to act less arrogantly and prideful, even asking to be introduced to Elizabeth's working class relations, the Gardiners, and engaging in conversation with them while on his estate at Pemberley. He also later confesses just how much Elizabeth's chastisements "tortured" him, making him want to amend his ways (Ch. 58). In the end, Darcy proves himself to be the most moral and noble character in the book as evidenced by how he rescues Elizabeth's and her family's reputation by bribing Wickham into marrying Lydia. Hence, we see that another part of Elizabeth's characterization is that her own sense of ethics and morality also inspire other characters to improve their ethical and moral conduct.