How can I write an essay on Harper Lee's To Kill a Mockingbird?

English teachers love To Kill a Mockingbird for a variety of reasons.  One reason is that the book is full of diverse themes.  For your essay, pick a theme of the book, and attempt to prove how that theme is displayed throughout the book.  You could pick friendship, racism, bravery, parenting, honesty, hypocrisy, guilt, and/or innocence.  Without knowing specific requirements for your essay, I recommend doing a thematic analysis of the book.  It's broad enough to give you lots of potential support, but narrow enough to keep you focused.  
I also recommend doing a compare and contrast essay on two characters from the book.  As a teacher, I prefer reading these because so many students go the theme route.  The character analysis is a nice change of pace for me.  You could compare and contrast Atticus and Aunt Alexandra.  You could also compare and contrast Jem and Scout.  They might live in the same house with the same father, but they definitely see the world differently.  Jem especially.  The book is a real coming of age story for that boy.  
You asked for some general writing tips to make your essay better.  I would like to give a few tips.  I don't know how long the essay needs to be, but it should be a minimum of five paragraphs.  The first paragraph must be an introductory paragraph that ends with your thesis statement.  The thesis statement needs to be a statement that you intend to prove correct.  It can't be a statement of fact, because there is nothing to prove.  
The final paragraph is your concluding paragraph.  Don't forget this paragraph.  It should do two things.
Remind the reader of the evidence that you just presented in the body of the essay.
Push your reader toward caring about your topic.
If the essay is a minimum of five paragraphs with an introduction and conclusion paragraph, that means the body must be a minimum of three paragraphs long.  If the topic can't be supported by at least three paragraphs, it's not a good topic.