I want to make an Apple ID, but someone has already used my email to create their own. How do I kick them out completely?
You'd have to try negotiate that one with Apple. 

I had the same issue with an Amazon account &amp; the only way I persuaded them was by actually being the only one who could reply to mail sent to that email address.

[BTW, it's still not easy as their first assumption would be that you hijacked/phished someone else's email account.]

After comments, moving more info into this answer...

The way I did it was to ask them to send a specific message, that only the receiver of the email could reply to.
The possible responses could come from me, him, or possibly both of us.
In fact, as the other user was never going to receive that mail, the response only came from me.

Because I'd already been receiving emails intended for him, I could also provide details of the contents of those, too, including a list of his recent purchases. 

Amazon [&amp; me, because of the incorrect email address] already had his snail mail address, so presumably they could sort out a new, correct, email address with him using that method - or put a big banner on his account saying 'get in touch'.
I don't know how they did sort it out because, of course, I couldn't get into his Amazon account, I was just getting his Amazon email.