(covers information from several alternate timelines)
"And there's a third one in our brig. I arrested him earlier today. But, don't worry. They'll all be reintegrated in time for the trial."
Encounters with Voyager
Braxton was first encountered by the USS Voyager in 2373, when the Aeon arrived in the Delta Quadrant through a temporal rift. Braxton claimed that Voyager had been responsible for the destruction of the Sol system in the 29th century, and had traveled back to prevent that occurrence by destroying Voyager. The Voyager crew was able to counter Braxton's attack, however, destabilizing his temporal rift, into which both ships were pulled. Both ships were subsequently sent back to Earth, the Aeon to 1967, and Voyager to 1996.
As the Aeon plummeted towards Earth, Braxton managed to execute an emergency beam out before crash landing in the High Sierras. He was not, however, able to retrieve his ship before it was discovered and stolen by a camper named Henry Starling. Over the next thirty years, Starling would utilize his primitive understanding of the timeship's technology to introduce new computer innovations to the public, constructing a corporate empire and initiating the computer revolution of the 20th century.
It was also during the same thirty years that he was able to develop a theory as how the disaster was caused: after Starling stole the timeship, he flew it into the future without recalibrating the temporal matrix which caused the explosion that destroyed the system.
Braxton followed Starling's movements, but was never able to reacquire the timeship. Eventually Starling became too powerful, and Braxton was ultimately unable to directly end Starling's threat. He became disillusioned with life among "post-industrial barbarians", and– despite his best efforts to explain his situation to various authorities, notably that of being a timeship captain from the future– he was eventually placed in a mental institution where they "filled [him] with primitive pharmaceuticals."
Following his freedom he lived homeless on the streets of Santa Monica, California. There he was still frequented by "damn social workers" offering their advice, and would often fall victim to "greedy people" who were "always trying to steal [his] things." His free time was often spent "putting up literature about the end of the world," and often prompting complaints from locals. Poor treatment at the hands of the authorities, whom he described as "quasi-Cardassian totalitarians", ultimately had left him paranoid and aggressive.
When Voyager eventually arrived in Earth's orbit, the ship detected the low frequency subspace readings from Braxton's combadge in the Los Angeles area, and eventually discovered him in his disheveled condition, pushing a shopping cart full of his possessions. After Captain Janeway and Commander Chakotay, from Voyager, followed him to the alley where he lived and confronted Braxton, he blamed Voyager for his predicament, and explained the paradox of their situation and his deduction of Starling's future doings.
During their encounter, Braxton was visited by Officer Sims of the Los Angeles Police Department, regarding the placement of this doomsday literature. When Janeway and Chakotay refused to acknowledge Braxton's plea with them to confirm his identity, he fled of foot from the scene with Sims in pursuit. (VOY: "Future's End")
A short time later, the crew was able to confirm Braxton's theories, and they destroyed the Aeon with a manually-launched photon torpedo just before Starling entered the temporal rift. The end result created a new timeline in which the explosion never occurred, but left the events of the past unchanged. Almost immediately after the collapse of the first Aeon's rift, a second rift opened and a second Aeon emerged after the Temporal Integrity Commission sent Braxton to correct Voyager's anomalous presence over 20th century Earth.
Initially surprised that the crew of Voyager immediately recognized him, and how, he explained that "he" never experienced "that" timeline. Despite Janeway's plea to return them to the Earth of their own time, as Voyager had been stranded in the Delta Quadrant for two years, Braxton escorted Voyager back to its proper place and time so as to uphold the Temporal Prime Directive. (VOY: "Future's End, Part II")
After his thirty-year exile on Earth, Braxton was forced to undergo extensive rehabilitation before he was allowed to return to duty, where after he was given command of the Relativity.
In that capacity, he recruited Seven of Nine from the year 2375 to a mission to save Voyager from destruction. Bringing Seven aboard the Relativity, he explained that a saboteur had placed a temporal disruptor aboard Voyager, and that they needed her to identify the individual responsible and prevent the disruptor from destroying the ship.
During the course of her investigation, Seven discovered that the saboteur was, in fact, Captain Braxton's counterpart from a future point in time. Braxton's counterpart explained that, in the upcoming few years of Braxton's life, he would be returned to rehabilitation due to suffering from temporal psychosis, and subsequently forced to retire. Braxton again blamed Janeway for these misfortunes, citing that due to the number of time travel events she caused that he was forced to repair, which included the temporal inversion in the Takara sector, as the cause of his condition. As such, he became determined to wipe Voyager from the timeline. Braxton's future counterpart attempted to convince Braxton to let him proceed and, in light of these revelations, Lieutenant Ducane, placed Braxton under arrest for crimes he was going to commit, and took control of the mission.
Seven was eventually successful in apprehending Braxton's counterpart, but Ducane was still tasked with repairing the damage done to the timeline due to their various temporal incursions. As Seven could not make another time jump without suffering from temporal psychosis herself, Ducane recruited Captain Janeway to complete the mission. In exchange for leniency, Braxton's future counterpart agreed to inform Ducane of the precise time and place where he planted the disruptor. Janeway was then sent to the year 2372, where she was able to apprehend Braxton before he was able to activate the disruptor.
This version of Braxton was subsequently transported back to the 29th century, resulting in the existence of no fewer than three versions of him being held in custody, although Ducane assured Janeway that all three of them would be reintegrated in time for him to stand trial. (VOY: "Relativity")
Braxton was initially described in the final draft of "Future's End" as being in his "early 40's, wearing a version of Starfleet uniform and combadge that is different and distinct." When he is later encountered in the 20th century, he was described as "a homeless man" [in his] "early 70's, ratty gray beard, longing hair, heavy Winter coat and snow boots, despite the heat. [...] Aged 30 years and looks like hell."