Hamilton Hill became Mayor of Gotham years before Thomas and Martha Wayne's murder in Crime Alley. He forged an alliance with the two and crime boss Carmine Falcone, running the politics whilst Falcone handled many of their operations and the Wayne's legitimized their money. Though challenged to his place as Mayor, Hamilton's opposition would frequently disappear, under suspicious circumstances. One of his previous rivals included Theodore Cobblepot, who apparently committed suicide part way into his campaign.
With Thomas, Hamilton was responsible for convicting people who refused their offers to Arkham, sending them insane with a psychogenic chemical. One of them was Esther Cobblepot, the last remaining "of age" Cobblepot, who refused to sell land that later became the site of Wayne Tower.
Eventually, Martha saw that Thomas was going too far and threatened to expose his crimes, before he did too much harm, worrying Hill. He hired one of Falcone's hitmen Joe Chill to assassinate Thomas and Martha and make it look like a mugging gone wrong. Chill did so as they and their son Bruce returned home from a movie showing at the Monarch Theater. However, Chill was caught, but Hamilton, and possibly Falcone, arranged for him to be murdered in prison.
Nearly twenty years after that event, Hamilton was approached by Theodore and Esther's son, Oswald, who wanted to expose Gotham's corruption and remove those who were corrupt from power. Either through blackmail or a willing alliance, Hamilton gave him evidence to prove the Wayne's ties to corruption. He also began compiling data to incriminate Carmine Falcone, either to hand over to Oswald or help boast his campaign as Mayor. Unknown to him, Penguin also planned to kill him, once he had served his purpose. At the same time, Hamilton was challenged for his seat of Mayor by District Attorney Harvey Dent.
At the beginning of the episode, Hill's office was attacked by the Children of Arkham and Catwoman, both of whom attempted to break into his vault and steal the data he had compiled on Falcone. However, the latter was successful whilst the mercenaries were caught by Batman. The next day, Hill authorized a police warrant to search Wayne Manor and collect documents related to Wayne Enterprises. This was either at the behest of Oswald or a smear tactic to ruin Dent's campaign.
Hill later called one of Penguin's men, demanding that he was let loose and that the debate was not attacked. Later, Bruce Wayne, following a confrontation with this man at the Stacked Deck Bar, came into position of this phone and heard a recorded call from Hill to this henchman. Deciding to confront Hill, he can choose to go either as himself or Batman.
Managing to charm Hill's secretary into letting him in, Bruce meets with Hill, who is practicing for that night's debate. When Hill questions Bruce on why he is here, the latter plays the recorded call made that day. Hill agrees to tell Bruce, if he promises to stop funding Dent's campaign.
If Bruce agrees to, Hill reveals how he, Thomas and Falcone took control of the city and gained property from those who disagreed. He also shares information on why Penguin is attacking Gotham. Despite Bruce's warnings, Hill claims that he is still in control. Before he leaves, Hill gives Bruce his father's cufflinks, hoping that the two will work together in the future. However, if Bruce refuses to, Hill will threaten him to not release the recording. He then orders security to escort him out the building.
Whilst on the call with his secretary, the power in Hill's office was suddenly cut. As he called out to her, Batman made his presence aware to Hill. The vigilante questioned Hill on his connection to the Penguin, even playing a recording on the call he had made. Hill tried to rebuke the conversation onto Wayne and, whilst he was distracted, called for security. Batman responded by dangling Hill from the balcony, demanding that Hill really told him the truth. He complied, telling him about Thomas Wayne's unlawful commission of Gotham citizens to Arkham Asylum to gain control of their land and wealth. Satisfied with what he had heard, Batman let Hill go and left as City Hall security guards appeared.
Hill was at the debate as it started, having prepared ahead of time for it. As it began, Penguin and his men attacked, murdering the moderator and taking the audience hostage. Bringing out a hostage to "host" the event, they forced her to inject Hill and Dent with the same chemical used by him and his partners to commit people to Arkham. Loosing control, Hill admitted to his darkest of thoughts, including his feelings about the citizens of the city and those in poverty. Penguin also played a clip of him and Wayne committing Esther Cobblepot to Arkham for refusing a deal they made to her.
As the GCPD tried to raid the debate, Batman attacked Penguin, but was brought into a fight with one of his men. As Penguin prepared to kill him, Hill confessed to having arranged Chill's assassination of Waynes, claiming that he had done so after believing Thomas Wayne had go too far. Penguin refused to spare him, claiming that his actions had not saved his mother's life. Though Dent tried to protect him, Cobblepot shot Hill dead, avenging his mother.
Following Hill's murder, Harvey Dent was unopposed and was sworn in as Mayor. Hill's reputation as Mayor was also tarnished, with his links to Falcone and the Waynes and activities with Arkham Asylum exposed by the Children of Arkham. Lady Arkham, who had helped organize his murder, also used his death, along with Falcone's, as an excuse to Batman for reasons why she had helped the city during their final confrontation. Batman himself can also see Hill's death as either a good thing, believing that he had received justice, or as a negative, believing that he and Falcone had not received the proper punishment.
Though he initially would appear to potential allies as friendly, Hill, much like Thomas Wayne and Carmine Falcone, was known to be ruthless and willing to use any means to get and keep control of Gotham. It seems he was also somewhat cowardly, often relying on his employees and allies to take care of problems or trying to shift the blame. Hill was also willing to betray others for his own personal gain, as he did with both Wayne and Falcone.
The precise relationship between Hill and Falcone is unknown, but it is known that they were allies and would work together to run Gotham, with Thomas Wayne. It is not known whether this became antagonistic after Hill arranged Thomas and Martha Wayne's deaths, but he would keep a drive on Falcone's activities, in the hopes of exposing it if he turned on him. It is believed that Hill was going to give this drive to Penguin and the Children of Arkham to expose Falcone and to keep himself safe from the group's wrath.
Thomas & Martha Wayne
The precise relationship between Hill and the Waynes is unknown, but Hill claimed that Thomas Wayne gave him a set of cufflinks with keys on them as a present, meaning that they would have been close. However, after learning Martha planned to expose the trio's actions, Hill arranged her and Thomas' murder in order to silence them. He may have targeted the latter, knowing that if Wayne found out about Hill's connection, he would have likely killed him in retaliation or worse.
Hill and Dent had a very poor relationship, due to his ties to various criminal organizations, allegations of corruption and the District Attorney threatening his position of power. Part of the reason why Hill had given the Children of Arkham evidence on the Waynes was because their son, Bruce, was sponsoring Harvey's campaign.
Before their first meeting, Hill was willing to hand over evidence on Thomas Wayne's activities, which would have lowered Bruce's reputation. If Bruce does meet Hill, he initially is pleased to see him, believing the billionaire wanted to have a close relationship, similar to that of his parents. Depending on whether Bruce agrees to pull funding from Dent's campaign or refuses to, Hill's relationship with the billionaire can be either positive, with him handing over Thomas Wayne's cufflinks and promising that they would do great things in the future, or antagonistic, with Hill threatening to humiliate him and Dent (and Gordon, if Bruce handed evidence over to the Lieutenant), should he hand over evidence that he had been working with the Children of Arkham.
On the other hand, Hill's relationship with Batman was one of fear, as he was aware of the vigilante and his methods. If Batman comes to talk with Hill, he will be willing to tell him whatever information he required, hoping that he would leave him be. However, Hill would also call for his security guards to save him from Batman, out of desperation and fear.
Oswald Cobblepot/The Penguin
Hill claims that Cobblepot had blackmailed him to retrieve information about Thomas Wayne and Carmine Falcone, something which the voicemails heard by Batman seem to back up. However, it is likely that he had worked with them, in the hopes forcing Bruce Wayne to cut off ties with Harvey Dent (who's campaign he was supporting). Hill claims that Cobblepot had been planning to attack the mayoral debate, but the criminal did so after Hill had handed over the information. When Penguin tried to kill him, Hill begged for his life, trying to claim responsibility for Thomas Wayne's death to stop his brutality, but Penguin, having seen Hill as the man who ordered his mother's drugging, murdered him, despite his pleas.