I'm quoting a book (To Kill A Mockingbird) and using a quote where the n-word is used. Later I'm going to have to read that out loud while my teacher also has a copy. Should I censor the word when reading or writing it?
As this is for a class, ask your teacher. If it was for a conference, it would be ask the chair and for a publication it would be ask the editor.
My preference would be to first determine if the quote provides something that a paraphrasing cannot. If I could paraphrase the jist, that is almost always my preference, even if the quote is not problematic. If you must quote, then you need to decide if you need to censor (or provide a disclaimer) about the language. Your style guide might help you here (but that is like asking the teacher/chair/editor).
As for presenting it when reading your paper, I would probably just omit the word with a pronounced pause, unless the word was key. If the word is key, then everyone will understand why it has to be said.
The book you are quoting from is very well-known, and it is unlikely that an audience would be unfamiliar with the language it uses. If you were to quote it verbatim, I doubt that this would be particularly controversial. Nevertheless, it should also be possible to quote the book passage in an altered form in a way that is not misleading to the audience. In any case, when giving a quotation from any source, to illustrate some point, it is acceptable to alter the quote so long as you have a valid reason (in the context of what you are saying) and you obey some basic rules. If you are going to alter a quotation to avoid offensive language, you need to do the following:
Proper attribution: Your alteration should be clearly attributed to you, so the audience knows that this is not a verbatim quote of the source. In written form this is accomplished by using the proper syntax for alterations (e.g., using ... to omit words or using [replacement] to replace words). In oral presentations you can get the same effect by showing the quote on the screen or by explicitly noting your change.
Don't mislead the audience: Ensure that the altered language does not mislead the audience on any material point relating to your presentation. In this post you refer to the "n-word" rather than saying the actual word, but we all know exactly what you mean, so it would probably not be misleading to use this as a substitute term. I recommend against replacing a particular epithet with a generalised [epithet] replacement, since the content of the epithet is relevant to the story.
With these points in mind, have a think about what you want to get across with your quote, and whether there is any particular necessity to include the racial epithet. If you want to omit the particular epithet, but make sure the audience knows which epithet was omitted, you could write something like this:
“Don’t you believe a word he says, Dill,” I said. “Calpurnia says that’s [n-word] talk.”
For a spoken presentation of the same thing, you should always ensure that you do not mislead the audience on the actual material. If you are using the textual replacement above, it would probably be worth noting at the start of your talk that you intend to do this, and that the book is uncensored.