Superman does not kill, but to save humanity he had to kill Zod who could not be trapped nor was he willing to surrender, Superman however is a good person so he is destroyed by pain for not having saved a life, even if it is that of a genocide supremacist.
The other reason is that Zod, rightly or wrongly, is the only Kryptonian still alive after Superman, Superman in fact, after discovering he was an alien, wanted to know what his people were like, and only Zod could tell him about Krypton. Superman then cries at the end as The Doctor cries at the master's death, both having become the last members of their race.

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