Joshua Said that he was expecting a courier, except it wasn't us. Apparently The other Courier was working for Caesar to kill Joshua seeing as how all the other assassins were killed.

Some of the speculation here is pretty extreme.
I think we can safely assume the following to be true --

Ulysses is human. Not a clone, not a cyborg, not an android, not an alien, not from the past or the future. Every major protagonist in this game (except Yes-man and House, to a degree) is a human being. As this is a story about people I doubt this will change.

Ulysses is not a blood relative of the Courier. Our characters can be one of four races; it's unlikely that there will be 4 Ulysses models in the game.

The relationship between Ulysses and the Courier will not be romantic. The game designers will not impose a sixual preference on the Courier.

Ulysses will not be firmly tied to any faction. I may be wrong about this (bearing in mind the 'join Elijah' option in Dead Money); it would be more desirable from a game play perspective to allow equal play time for members of each faction. Also, as a DLC, the story must be playable at any point in the MQ -- it cannot rely on faction loyalty by the player, either.

Okay, here's my speculation.

Ulysses once was a Bighorner rancher in the Mojave. At some point, perhaps after an incident involving the Courier, he leaves to travel the world. Dissatisfied with the factions vying for the Mojave, perhaps after working with one or more (the Legion connection?), he wears the flag of a dead nation on his back.
He explores the Big Empty, performing fairly heroically. He finds a weapon there.
He test detonates this weapon (created under an old world flag) at the spot that will become the Divide [At a strategic point that disrupts the NCR without totally crippling them, maintaining a precarious balance of power].
He maintains a presence in the legion throughout this time, gaining reputation (deserved?) as a Frumentarii assassin.
In Primm, he discovers that the Courier is still alive (was he expecting the Courier to be dead on general principles or did a specific incident lead him to this belief? Remember the 'two bullets' controversy -- was Benny the first person to fail to execute our hard-headed Courier?) and refuses the Platinum Chip delivery. Was this really an act of revenge, or did he want the Courier to get 'sorted out' mentally about the state of the Mojave? Or did he refuse delivery because he feared being recognized by House?
In Lonesome Road (referring to Ulysses's journey?), the Courier is contacted by Ulysses and summoned to the Divide. I believe that Ulysses will attempt to hire or secure the loyalty of the Courier, as a sort of mini-faction, to help him use the Divide Weapon against the enemies of the Mojave. It's a given that, save for a Dead Money-style early ending, the Courier must refuse, either because of the nature of the weapon or because Ulysses is, in fact, unhinged.

Here's an alternate theory -- perhaps Ulysses is just a mercenary and wants to sell the divide weapon -- there may be representatives of each faction appearing in the DLC. That could be fun. Not sure what the Courier's role would be if that is the case.