Should I use 'whoever' or 'whomever': "I will kill ___ despises me." ?

I know this sentence is a little awkward. Bear with me.

"I will kill whomever I despise." -- This one feels correct. However...

"I will kill whoever despises me." -- Is this right? Would this one also be whomever? Or is 'whoever' correct here?

Both sentences are grammatically correct in that you've used the correct whoever/whomever in both. Break each sentence into two:

I will kill him. I despise him. [him + him = whomever]

I will kill him. He despises me. [him + he = whoever]

But the meaning differs between your two sentences. In the first you are saying you will kill people you despise. In the second you are saying you will kill people who despise you. Which is it?

Also, if you're a high school student in the US I wouldn't let my teacher see this.

Well, if we compare the uses of the root words, "who", and "whom", we can easily determine when to use "whoever" and when to use "whomever". "Who" is used when the person in question is the one acting. Eg. "Who ate this cake?" (someone has committed the act of eating the cake.) "Whom", on the other hand, is used when speaking of someone who is not acting. "To whom shall I grant the privilege of eating this cake?". The "whom" is used to describe a person who is not acting. So, "whomever" can be used when speaking about someone not committing an action, and "whoever" when that person is committing an action. For your first sentence, you are speaking of someone who is not committing an action, therefore, "whomever". The second is "whoever", because they are committing an action (hating you). In short, both of your sentences are correct.