I think I've found out the Arbiter's real name, and I can fill in a correct bio for whoever's playing him if needed. Actually, I'm just going to post all the information I've found.
Role
The rank of Arbiter is bestowed upon a Covenant Elite by the High Prophets during a time of extraordinary crisis. The Arbiter acts as an agent of the Prophets, going on missions specifically dictated by the Prophets to resolve whatever difficulties the Covenant is experiencing. During incidents such as the Taming of the Hunters, the Grunt Rebellion, and most recently, the threat of heresy, and everything up to, but not including, the Covenant Civil War, the Arbiter of the time has emerged to lead the Covenant to victory. Many, especially Grunts, see the Arbiter as their savior or protector, presumably because only Elites of extraordinary skill and leadership attain the rank. The Arbiter is commonly referred to as "The Will of the Prophets" or similarly "The Blade of the Prophets". The Arbiter wears ancient armor that differs from that ordinarily worn by Covenant Elites.
Every Arbiter has been martyred in the undertaking of their momentous tasks. The corpses (or perhaps only memorials of some sort) are housed in identical caskets stacked upon each other in the great Mausoleum of the Arbiter, located on High Charity. In the center of this room a floating pod contains the sacred armor of the Arbiter, highly decorative and fully functional (although some subsystems are out-dated, such as the active camouflage) despite its apparent age. The mausoleum houses 168 visible caskets (and more may be set underground), reaching high into the air, and references of Arbiters settling the Grunt Rebellion and aiding the Taming of the Hunters would indicate that the office has existed for quite some time - at least before the Covenant was fully formed and its castes defined. The Arbiter can only be an Elite (the characteristic armor is tailored to Sangheili anatomy), as they are traditionally seen as the guardians and protectors of the Covenant.
As a member of the Covenant, it is believed that the Arbiter is held in the capacity of maintaining the integrity of the Covenant, and his creation is necessitated only in momentous events like those mentioned above. Of course, he also serves as a political tool, an embodiment of the Hierarchs' power as well as an example of true faith, a warrior-martyr showing loyalty at times of division.
History
The Arbiter in Halo 2 was previously an Elite Supreme Commander, having commanded the Fleet of Particular Justice, the fleet that destroyed Reach and followed the Pillar of Autumn to Alpha Halo in Halo Combat Evolved. The reason the ship was even allowed to land in the first place was due to the risk it might pose to the ring; a minor Prophet onboard one of the first ships to engage the Pillar of Autumn refused to allow the ship to be destroyed with plasma torpedoes, instead ordering it to be captured by boarding parties. This error allowed the Pillar of Autumn to land on Halo's surface, and events spiraled out of control, leading to the Master Chief and Cortana detonating the Pillar of Autumn's fusion drive and destroying the ring. In Halo: First Strike, it is also hinted that he was responsible for the loss of the Ascendant Justice.
Looking for a scapegoat for the destruction of the "Sacred Ring", and unable to blame the Prophet, the Covenant High Council turned to the Supreme Commander, branding him a heretic and stripping him of his rank. To signify this, he is branded with the Mark of Shame over his left pectoral by Tartarus in front of a large crowd. Though his execution (hung by his entrails and his corpse paraded through the city as an example) was soon to follow, he was spared by the High Prophets.
During a meeting with the Prophets of Truth and Mercy within the Mausoleum of the Arbiter, they reveal that they were quite aware that the council only wanted a scapegoat, and that branding the Supreme Commander a heretic was the only way to quell their malcontent. Rather than have him slain outright, the Prophets of Truth and Mercy gave him a chance to regain his lost honor and serve the Covenant again; he would become the Arbiter and be set loose against the true 'heretics', lead by one Sesa 'Refumee, and quell their uprising before it caused further division amongst the Covenant. As they explain to him, the council will get their corpse one way or another, as he is likely to die in the face of the overwhelming odds they are pitting him against.
Like the Arbiters before him, the Prophets expected the Arbiter to die early in his tour of duty, completing his apparent "death sentence", but he survived both the destruction of the heretics and the storm that followed. Thus, they send him on another suicidal mission, this time to retrieve the "Sacred Icon" (Halo's Index) from the Library on Delta Halo. To be sure it would kill him, they secretly order Brute Chieftain Tartarus to do so in the unlikely event that he survives the attempt. When the Arbiter succeeds, Tartarus throws him down the central shaft of the Library. Though Tartarus is convinced that he succeeded, the Flood Intelligence Gravemind rescues the Arbiter for his own purposes.
During the game, the Prophets slowly begin to replace the Elites with Brutes, a species that has shown to be both stronger, more resilient, and more blindly devoted to the commands of the Prophets than Elites. Once there are enough Brutes in place, they instigate a bloody genocide, wiping out the Elites. This sparks a massive civil war, disguised as a Brute insurrection to the Elites and their allies, and as an Elite revolt to the other Covenant factions (excluding the Brutes). On one side, there are the loyalists: the Brutes, the Drones, and the Jackals, covertly sanctioned by the Prophets. On the other, the Elites lead the Hunters and the Grunts.
The Arbiter, after meeting his human counterpart, the Master Chief, while being held captive by the Gravemind, discovers the Brutes' massacre of a number of his fellow Elites and receives word that the Elite High Council members have been murdered. He joins forces with the separatists and becomes a central figure in the Covenant Civil War.
In order to keep Tartarus and his Brutes from activating Halo, the Arbiter forms an uneasy alliance with their enemies, the human United Nations Space Command, to stop the mutual threat. At the end of the game, the Arbiter confronts Tartarus in the control room of the installation. While he had originally, and perhaps stubbornly, refused to believe the truth about Halo, the common belief of the heretics, Master Chief, the Gravemind, and Sergeant Johnson that Halo would bring devastation finally convinces him to ask 343 Guilty Spark the truth about the rings. Though the Arbiter accepts this truth, Tartarus does not, activating the weapon anyway. The Arbiter triumphs in killing Tartarus and the humans stop the activation of Delta Halo, though the others are armed as a result. In the final scene, Guilty Spark explains to the Arbiter and the humans that the Halo network can be activated from the Ark, but when asked the location of it by the Arbiter, the scene ends.
I will not explain what happens in Halo 3, as this isn't the time for it yet. That's all the information I found.