Copied from TA:
King of the Hill is an online mode, so you have to go to:
"Online" - "King of the Hill" - "Survior".
Then you can either search for a public match or play together with a friend in a private match.
It doesn't matter which one you pick, just win a single match in either public or private and the trophy is yours.
But keep in mind, that playing a private match with only one buddy will not allow you to give respect points, as only the crowd and not the fighters can award those.