"I will finish you" means that the speaker is going to be a destructive force for the listener.
Context is important. It could mean "I will kill you."
But in if it is said at work, it could mean "I will make sure you are fired from your job and that no one else will ever hire you again. You will be jobless."
It could mean "I will take away all of your money."
If it is said in a social setting, it could mean "I will convince your spouse to divorce you and all of your friends to abandon you."
It is not a nice thing to say.