They made a bet with eachother: "Whoever takes a bullet to the head and lives wins." The courier won, so Ulysess is pissed.
This one sounds pretty legit.
The only clues we have to go off of are:
1) Ulysess seemed surprised to learn the Courier is alive.
2) Ulysess knew the Platinum Chip job was bad luck.
3) The devs have said their relationship will surprised us.
4) Ulysess said "People are like Couriers, never really understanding the impact of the message they're delivering."
5) Ulysess' card in the special edition deck is a club. Clubs are Caesar's suit.
6) Wolfhorn ranch is assumed to belong to Ulysess due to some of the foreign translations of the location's name. If you visit the Ranch and find the grave mentioned there, it has a burnt up book and a harmonica on top of it. (I'mma laugh if Ulysses turns into a "Once upon a Time in the West" reference)
My guess is the Courier probably unknowingly and inadvertedly got one of Ulysess' loved ones killed, and since then Ulysess has tried multiple times to kill him inadvertedly. I'm probably completely wrong though.