Usually, it'd go "I don't want to be like you. Ever." It's a word you hear in very informal conversations, usually used for emphasis on distaste. I hear it frequently from teenagers at my local high school. (Don't mean to stereotype this)
To go slightly more formal, you could also say "I don't ever want to be like you". Same meaning, less intense, and more widely used.
I have a question. Can "ever" go in the final position of a sentence?
like: I don't want to be like you ever
Yes, there are some places where you can do this. For example, the "Lord's Prayer": "For thine is the kingdom, the power, and the glory, for ever and ever."
But most instances like this would read "forever". Your sentence is acceptable, but not the most common way of saying it.