so another me from alternate reality comes to meet me and I had to kill him for some reasons and I'm caught by the police. Following scenario takes place in US and our world is unaware of alternate realities and science is not yet evolved travel between the different realities
If the world is not aware of alternate realities (and possibly even if it is), then your alternate self is just another person who happens to look like you, and you have just committed murder. You will be charged with the same thing that any other murderer would be charged with, which may vary depending on the circumstances (self defense, etc.)
Determining their citizenship may be interesting (since their ID is the same as yours) but the most lenient things could get for you is if they are judged to be an unregistered alien who stole your identity. And I'm pretty sure that killing an illegal alien is still illegal in the US.
Details might actually vary from jurisdiction to jurisdiction, but murder is generally defined as the killing of a human being. If there is no contest on the "killing" part, could the defense lawyer of your character question the "human being"?
I believe that US law defines human beings as homo sapiens, thereby kicking the ball to the biologists. I don't think that "planet of origin" matters for the biological species definition -- surely not even a lawyer would argue that a Mars-born human is not human for that reason.
Perhaps the nature of the victim might help the character with an insanity defense, claiming panic at the sudden arrival of a perfect double.
Prefacing this with a disclaimer: I am not a lawyer, and this should not be taken as legal advice. The law may vary slightly from state to state, so it may not be fully accurate in the reader's location. With that said, in the US in general:
Murder requires premeditation. If there is evidence that the killing was planned, then you could be charged with murder. You see your neighbor having sex with your wife, and in a fit of rage, you grab your gun and run next door to shoot them? That's murder, because grabbing the gun showed premeditation.
Felony Murder requires that the person was committing a felony crime was being committed, and someone died as a result. If you rob a bank, and a police officer pulls out their gun and shoots at you, misses, and kills a bystander, you get charged with felony murder because that person wouldn't have died if you hadn't tried to rob the bank.
Manslaughter requires that the killing possessed intent to kill, that the death was the result of negligence, or that the death occurred during the commission of a misdemeanor crime. Killing someone because you were driving unsafely would be manslaughter, as would pulling out the gun you habitually carry and shooting someone in the heat of passion. If someone tripped and fell, and died from hitting their head on the pavement while they were chasing you because you stole their wallet, that would be manslaughter too.
Self Defense isn't a crime, but it requires the person to have been in genuine fear that the person they're defending themselves from might be a threat to their lives. Home invasions count, since they're often violent, and the law doesn't require you to actually find out what the person's motive actually is.
I don't think that the deceased was a copy of the murderer from an alternate universe would matter.
If found with the body you are getting arrested, what happens after could be interesting though, it may be possible to argue that since the victim is in fact you and you are on the stand that no crime can possibly have been committed. I wouldn't want to try to argue it as a lawyer and I really wouldn't want to be reliant on such a defense as the accused. I'd think one to be better off using an insanity defense, you thought the deceased was you, which is impossible, and killed them in a fit of... confusion, narcissism, something else maybe? You are either going to jail or a secure unit for the criminally insane.
Alternatively if you can successfully dispose of the body in such a way as to divert initial suspicion as to the identity of the killer confusion over the identity of the victim, since the victim is identifiable as very definitely you and you are still running around happy and healthy if somewhat confused, may be sufficient to have the investigation stall and eventually collapse.