EDIT: It appears that playing in a Private Match does NOT work anymore. Tried boosting with someone and it did not unlock after 5 match wins.
King of the Hill is an online mode, so you have to go to:
"Online" - "Rooms" then press
to Create a Room. Name it something innocuous like TT KOTH so your friends will recognize it's yours. Alternatively, you can simply enter an empty room with a friend and proceed from there.
Once you're all in, highlight a friend and press
and highlight KOTH. Press
again and press
or
to add or remove slots to the amount of people you're hosting. Once done, press
and you'll begin.
This can also tie in with Respected Fighter as you'll give each other respect points and that only works in public matches:
You can begin taking turns, or just do 5 in a row and then switch. Not too difficult. If a random joins, you can back out and go into a different room that's empty or create another.