"That is, if he will only kill whom he loves, he believes that he has the right to help whom he hates"
Strictly speaking, "whom" is correct. In your sentence, there are two relative clauses (I've underlined them) and in each case "whom" is the object of the verb:
"whom he loves" = "he loves some person".
"whom he hates" = "he hates some person".
When a relative pronoun is object of a verb, as it is in your example, it should strictly be in the objective case which is "whom", not "who". Another way you can tell is by replacing "some person" with a personal pronoun, in which case objective "him" (not subjective "he") must be correct:
"whom he loves" = "he loves him /he"
By contrast, if it were the subject of a relative clause, as in "The Lord loves even those people who hate him", then the subjective case "who" is correct because it is the subject of the relative clause:
"who hate him" = "those people hate him".
Having said all that, "whom" is rather formal in your example, and no one will take much notice if you use "who" instead in informal speech, but don't use it in formal writing. I would deduct a mark from a student who wrongly used "who" in an exam.
Incidentally, in examples like yours, "whomever" is often preferable to "whom".