Background
Jack's father died when he was young and his mother left him in the 'care' of her mother, Jack's grandmother, who was (at a minimum) physically abusive towards her grandson. The emotional trauma from these events was strong enough that as an adult, Jack had his grandmother murdered, and no doubt contributed towards his mental decline.
Before the opening of the first Vault
Later in life Jack married and had a daughter, Angel, one of the universe's rare Sirens. During Angel's childhood, Jack was surprised to learn Angel had the power to control technology, which ultimately made her a target for bandits and other kidnappers; a fate that would ultimately come to pass in a standoff between a kidnapper and Jack. Jack tried to plead for Angel's return to safety from said man who adamantly refused, and was caught off guard when Angel, in a fit of rage, took over a turret and killed both the kidnapper and her mother (by accident), which Jack immediately reacted to with sorrow and blame.
During Angel's youth, he trapped her in Control Core Angel to use her to control Pandora's ECHOnet and to open the vault of The Warrior. In the Borderlands 2 story mission Data Mining, Handsome Jack says that "I know you think I'm a monster. You think I enslaved Angel. But you didn't see what she did to her mother. I had to restrain Angel's power. You get that? I had to." The seeming implication is that Angel caused her mother's disappearance.
However, this interpretation contradicts the journals in Get to Know Jack. While it is true that the third journal says that Jack's wife disappeared, it also says that Jack's wife suggested that "the Angel" be shut down. For there to be something to shut down, Control Core Angel must have already been active at that time, and therefore Jack's wife's disappearance could not have been the reason for the creation of Control Core Angel. Nor could whatever Angel "did to her mother"--if indeed anything happened at all--have been serious enough for her mother to agree with restraining Angel.
A conversation in Tales from the Borderlands explains this discrepancy by clarifying that Jack remarried at some point after the death of Angel's mother and that it was his second wife who wanted the "Angel Project" shut down and who left him over it. When an employee later brings this wife and her departure up to Jack, he angrily strangles him to death.
After the opening of the first Vault
BLTPS Jack Face Scarred
Jack after being scarred by Lilith
During the events of Borderlands: The Pre-Sequel, it is revealed that Jack's initial plan was to manipulate the Vault Hunters into opening The Vault to release The Destroyer so he could obtain it for himself. After The Destroyer's defeat, he used its eye as the generator for a superweapon dubbed "The Eye of Helios", capable of destroying entire settlements. Jack planned to use the weapon to eliminate entire bandit camps and other undesirables en masse on Pandora to eventually civilize the planet. Unfortunately, the Eye was captured by Colonel Zarpedon, who promptly turned it on Pandora's moon, Elpis in an effort to prevent its Vault from being found by Hyperion, nearly leading to the moon's destruction. Jack managed to retake the Eye from Zarpedon with the aid of a group of Vault Hunters he had hired, along with assistance from Lilith, Roland, and Moxxi, who then betrayed Jack by destroying the Eye, not wanting him to have that much power.
Once the Eye was destroyed, Jack required a new weapon of power. He managed to open the Vault on Elpis, which housed an object shaped like the eponymous Vault symbol that contained the knowledge of the Eridians. As Jack grabbed the object, he began seeing visions of The Warrior, the Vault it was sealed in, and the Eridium buried through the planet. Lilith then appeared from a portal and destroyed the object, causing a blast of energy that destroyed Jack's left eye and branded his face with the Vault symbol.
After the opening of the Elpis Vault
After having his trademark mask grafted to his face, Jack proceeded to murder the President of the Hyperion Corporation, Harold Tassiter, and declare himself the new President. From this point forward, he began calling himself "Handsome Jack".
Shortly after killing Tassiter, Jack discovered evidence of the H-Source, a collection of Hyperion technological secrets, buried in the experimental "Fragtrap" Claptrap's system. After the Vault Hunters extracted the H-Source, Jack used one of the modules to remotely shut down all robots of the Claptrap line, and he personally shot the Fragtrap prototype.
Jack then began focusing all his efforts on obtaining the Warrior and exacting his revenge on Pandora and the Vault Hunters. He used his daughter Angel to gain full surveillance of Pandora and control of the ECHOnet system. He then established mining operations to dig up all the Eridium on the planet, which would be used (among other, more gruesome experiments) to create Hyperion's proprietary E-Tech line of weapons.
Jack also ordered Hyperion to enact corporate and military control over Pandora. With help from an informant, Jack located New Haven and destroyed the settlement with his cyborg subordinate Wilhelm. When New Haven's administer Helena Pierce tried to escape with survivors, Jack captured Pierce and personally executed her before having Wilhelm kill everybody else. He later tracked down the Vault Key and its owner Dr. Patricia Tannis, who was tortured into giving up the object. Jack then had Angel artificially charge the Vault Key by pumping her body with a constant supply of Eridium.
Jack intro
Handsome Jack's Intro
Handsome Jack sought to lure out Vault Hunters to his sphere of influence by displaying a need for them, then left them for dead in the Windshear Waste. He constantly communicates with them through the ECHOnet, taunting them, threatening them, and otherwise manipulating them. His building of Opportunity and the drilling into Pandora's core is no secret to the citizens of Sanctuary and Overlook, but he still keeps a leader-like façade when addressing the citizens of Hyperion-settled villages.
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Involvement
Following Angel's death, Jack makes his first physical appearance in the story and kills Roland. Lilith attempts to attack him, but Jack throws a Siren power-restraining collar onto her neck and captures her to replace his recently deceased daughter as a catalyst to charge the Vault key. He rescinds the bounty on the Vault Hunters, vowing to exact vengeance himself for Angel's death, and threatening to hunt down anyone who kills the Vault Hunters before he does.
His final appearance is at the Vault of the Warrior and is fought as one of the two final bosses. After being defeated by the Vault Hunters, Jack takes the already-charged Vault Key and uses it to summon and control the Warrior. The Vault Hunters defeat it as well, thus ruining his plans for dominion over Pandora. Jack angrily chastises them for stopping him from bringing his vision of order to the lawless planet and is then killed by Lilith or the Vault Hunters.
Tales from the Borderlands
When Rhys attempts to use Professor Nakayama's ID drive to track the money he lost, he collapses onto the floor, and hears a mysterious voice claiming to be disappointed at how Rhys is trying to follow in "[his] footsteps". Later, while infiltrating the abandoned Atlas warehouse, Rhys tries to get the lift operational but fails as his system suddenly runs into glitches. The mysterious voice once again taunts him. After Bossanova is killed by Zer0, Rhys and the gang (Fiona, Vaughn, and Sasha) discover a secret room containing Atlas technology, including information about a "Gortys Project". The voice then begins to get clearer inside Rhys' head and is then revealed to be the voice of Handsome Jack (the same AI Nakayama produced in Borderlands: The Pre-Sequel in case of Jack's death), who then appears behind him as a hologram. Rhys initially suspects Jack is indeed just a hologram, though Jack soon shows he is self-aware and active. Initially, the Jack AI does not realize nor believe that he is dead. He tells Rhys to bring him back to Helios so he can be transferred into a new body, allowing them to run Hyperion together (but "mostly [Jack]" will control it). Jack later upgrades Rhys' Echo-Eye implant to give him the ability to hack into machinery and control it at his will. When Rhys and Fiona finally discover the Gortys Project's location, they are attacked by numerous sentry drones, which Rhys can use for his assistance against Vasquez and August. If so, Jack will take control of them by merging his consciousness into Rhys'.
The Jack AI then forces Rhys to assume position as Hyperion's CEO and reveals that it intends to create a Jack army out of him and many others. Rhys, in an attempt to stop the Jack AI, shuts off Helios' power and the entire station collides with Pandora, destroying it. However, the Jack AI still survives the ordeal and attempts to kill Rhys with his own cybernetics, forcing him to tear them off. Rhys has the option of completely destroying the ECHO eye to eliminate the Jack AI once and for all, or spare it, leaving him alone in an endless void.
Moxxi's Heist of the Handsome Jackpot
During Borderlands 3, it is revealed that Jack had created a large casino simply named "The Handsome Jackpot". Through audio logs and discussions with those trapped aboard, a picture of the casino can be discerned.
The Handsome Jackpot was a scam, designed by Jack to bleed customers dry of their money. When they arrived, they'd often find themselves paying for even minor things aboard the Jackpot, and would also find the machines rigged to help pay out less. Additionally, Jack would find ways to impound vehicles and escape options from the Jackpot which would force individuals to essentially live on the Jackpot and spend everything they owned. Jack would keep a gigantic hologram of himself floating above the Spendopticon, would have his Jack doppelgangers walking about the casino for him, and had hologram devices located about the Spendopticon and other locations to explain location and attractions. Some of the holograms don't explain much, however, and a few imply that by the time Jack had created the Jackpot, he had already been suffering from serious moral myopia. One hologram doesn't even relate to the Jackpot, instead seemingly recorded while Jack yelled at an employee to bring him pretzels, telling said employee he will die whether he brings him the pretzels or not as he doesn't actually want them.
Jack would spend most of his time on the Jackpot in the VIP Tower, a place that is off-limits via a series of powerful turrets. The only other assured person aboard the VIP Tower was Pretty Boy, hired on possibly as a financial adviser. Several ECHO logs can be found that show Pretty Boy attempting to give Jack financial information regarding the Jackpot, however, Jack refused to listen. He instead spent his time mocking Pretty Boy and insulting his appearance, calling him funny and nicknaming him "Pretty Boy". When Pretty Boy showed resistance to the title, Jack told him he would be fired if he didn't agree to the name. He also created the Jackpot battle suit which Pretty Boy would later use.
After Jack's death on Pandora, a security protocol on the Jackpot was initiated. It's possible this protocol was intended to trap whoever killed Jack aboard the Jackpot, but instead was activated as a byproduct of his death elsewhere, trapping innocent individuals aboard. The patrons of the casino became trapped as the security refused to allow them access to their ships or means of escape. The Jack doppelgangers attempted to help, but as would later be explained by Timothy (after he began using his real name again), many of the Doppelgangers were killed by proxy due to anger leveled at Jack. The casino fell to ruin and was overrun with insane former patrons known now as Looters. Pretty Boy, still trapped in the VIP Towers, wanted to deactivate the shutdown and gain control of the Loader construction facility below the Jackpot using one of the "Winning Hand" devices grafted to the Jack doppelgangers. With Timothy the only one left, he began actively fleeing anybody to avoid being killed for looking like Handsome Jack. It's also revealed that Jack changed the bomb in Timothy's face to explode if he ever left the Jackpot, trapping him aboard either way.
By the time the Vault Hunters of Borderlands 3 arrive, the casino has been locked down for years, and Moxxi, in a form of revenge against Jack for their poor relationship and a bad breakup, wants the casino for herself. During their time on the Jackpot, the Vault Hunters constantly find information and material related to Handsome Jack, and characters reference him frequently despite his death. It's also during their time on the Jackpot they can find evidence that Jack never truly let go of Moxxi in thought. He has a knockoff bar called "Foxxi's", and in his control tower, one can find a picture of Angel on one side of the room, and a picture of Moxxi on the other