What does the killing in "Killing Eve" mean?

There is a TV show named "Killing Eve" and its Chinese translation basically means Eve is killed or to kill Eve. But I wonder whether killing can be an adjective here to describe the state or action of Eve. And it means Eve is killing others. 1. Eve is killed. / To kill Eve. 2. Eve is killing other people. Which one is right? Thanks a lot.

"Killing" can be an adjective or present participle

You are correct; "killing" can be used as an adjective (or as a present participle—in many contexts, it is difficult to distinguish participles from adjectives) to describe something or someone that kills. An example is the phrase killing frost, which refers to "frost that kills".

So the English phrase "Killing Eve" is technically ambiguous.

But I would say that the interpretation where "killing" is a gerund, and "Eve" is the object of the gerund (the one who is killed) strikes me as more natural, so it makes sense to me that the Chinese translation is based on that parse of the title.

Either interpretation would be "correct" in terms of grammar

There is no way to use the rules of English grammar to determine which interpretation of "Killing Eve" is "right". You could ask the person who came up with the title to tell you the originally intended interpretation, but not everyone would agree that this should be considered the only correct interpretation of the title. Translators often don't have access to this information. (See the answers to this Movies & TV SE question about the translation of the ambiguous movie title "The Last Jedi".)