I have a story I'm writing which has a villain that, in order to make him more human/developed, I gave him a relatable, tragic and/or disturbing life story, gave him plenty of reasons to be who he is and do what he does (understandable or almost), and made him the hero of his own story.
For story reasons, he needs to be killed, and also kind of deserves this because of his deeds. "Kind of" because his reasons and story makes very blurred the line between evil and simple cause and effect, so it's not that simple to just say that he did what he did "because he is evil". However, I don't have the intention of making his death an emotional moment, because he is supposed to be seen as unlikable, an unlikable character that has a point, a victim that becomes a perpetrator. The hero, although having more positives than negatives, isn't a saint either, having at least a bit of the blame (or at least in the villain's vision).
And here comes the problem. This story is the first of a trilogy where this hero is the protagonist. I think that the hero needs to be likable the most possible to make the reader urge to read more about him. Previously, I asked questions regarding problems with this same hero's likability. I was having these problems because I was incorrectly characterizing/developing this character. Now that it seems that I got it right, and now that the villain is also well developed, now the villain is making the hero seem unlikable for killing him. If this was a one-shot fiction, such ambiguity of who is the real hero would be great, it would be amazing, but the problem is that there's still two sequels with this protagonist to go, and I can't simply discard them just because of some mere details in the first title.
So how can I make the villain sympathetic without affecting the hero's likability for killing him?
I think I forgot to include in my question a very important detail of my problem, and I need to clarify what I meant in my question. By "hero of his own story" I really mean "hero of his own story", as the villain is a narcissist. He's not doing any good to other people, the only one benefited with his actions is his ego. So it's not a case of "doing the right thing to other people the wrong way", but instead it's "doing the wrong thing the wrong way towards other people, having as final intended effect the satisfaction of his own ego". The villain sees no reason why he doesn't have the complete right to do what he wants, and the story shows that he indeed is not totally wrong, but the means he used are too extreme for "solving" his problem.
After reading the answers and thinking about this matter, it downed on me that actually the villain isn't as sympathetic and the hero isn't as "monstrous" as I was thinking. I think I got so much in the villain's head that I started to think just like him, seeing the villain as a hero or someone worthy of sympathy, and seeing the hero as a villain or someone unlikeable. I think that's why I was having this problem. Now I don't see the villain as positively as when I asked this question, especially because what he does is wrong no matter his reasons or what caused him to be that way, so it makes no sense to see the hero as unlikable for him simply doing what is necessary.
Also, Dan J.'s and Seserous' answers were spot-on on this matter, because even though the hero just does what needs to be done, he did that coldly and with wrath, that's another reason why it was leaving a negative impression on him for killing the villain.