Arkham Asylum was an ailing mental hospital and prison located outside of Gotham City. Once a well regarded institute for the mentally ill, the building fell into disrepair, due to the actions of Thomas Wayne, Hamilton Hill and Carmine Falcone. Arkham would become one of the largest parts in the Children of Arkham's plans, particularly against Bruce Wayne/Batman.
Founded in November of 1921, Arkham Asylum was run by the Arkham family for many generations. However, around either the late 1980s or early 1990s, the once state of the art mental hospital would become part of the operations of Thomas Wayne, Carmine Falcone and Mayor Hamilton Hill. Together, the trio would commit innocent people who stood in their way to the asylum, using a psychoactive agent to induce psychosis. With this, they would commit hundreds of people to Arkham, clearing them to take over much of Gotham City. However, there were some inmates that the trio would release to gain further influence, even if they still need to stay at the facility.
Eventually, the current generation of the Arkham family would decide to blow the whistle on their activities, having either decided to no longer stand by or discovered the information themselves. However, Wayne would arrange their assassination to ensure they would not expose the truth, leading to them taking over the Asylum. After Thomas' death at the hands of a hitman, Hill would stop his activities, but asked Falcone to stockpile the chemical for future use.
Over the next twenty years, Arkham would only fall deeper into disregard and mismanagement. Many of its inmates' condition seemed to get worse and its staff became more disillusioned. Inmates escaping become a normal occurrence and the orderlies were bribed regularly. Despite this, some staff, including Head of Psychiatry Dr Joan Leland, would stay optimistic and try to help the inmates as much as they could. At some point, a political group called the Children of Arkham formed to raise awareness of Wayne, Falcone and Hill's actions within the asylum, but fell on deaf ears.
Over the years, multiple inmates were committed to the asylum, including serial killer Victor Zsasz and ventriloquist Arnold Wesker. One particular inmate was turned up without staff knowing. As they had no record, they took to calling him "John Doe". Harleen Quinzel was also employed at Arkham and was a therapist for John, though left some time before the Children's attacks on Gotham.
Nearly twenty years after his murder, Thomas Wayne's son Bruce and District Attorney Harvey Dent decided to replace Arkham with a new mental hospital. Backed by Wayne Enterprises board members, including Robert and Regina Zellerbach, the new mental hospital was approved by the city council.
Outside Arkham Asylum, Bruce and Harvey unveiled their plans to replace the institute with the hospital and its name, the "Thomas and Martha Wayne Mental Health Facility". During the press conference, Bruce revealed plans to either leave the original building standing or tear it down, when asked about its fate by Vicki Vale. However, the conferenced quickly turned to questioning about recently unearthed information about his parents' ties to Falcone and organized crime, prompting him to leave.
During their attack on the Monarch Theatre, the Children of Arkham, now a terrorist cell, revealed a video clip of Esther Cobblepot being committed to the asylum unlawfully. They also, using either a mole in Wayne Enterprises or files they had received from Hill, accessed Arkham's files to gather evidence of Wayne's actions and made them public. Children of Arkham member Roland Desmond was committed to Arkham after being defeated by Batman, though it's never specified why.
A week after the debate, Bruce was committed to Arkham Asylum by Harvey Dent, due to him attacking Oswald Cobblepot at press conference. During his time there, he met "John Doe", who revealed Lady Arkham's past to him and created a riot to help him escape. The result of this was successful, due to either Wayne's lawyers being able to convince staff to release him or Bruce offering to testify for Lt Gordon. After defeating Harvey Dent at Wayne Manor, Batman can recommend that the Mayor be committed to Arkham, hoping it can help him recover.
After both Cobblepot and Dent were defeated by either Wayne or Batman, the Children of Arkham broke into the asylum, planning to release its inmates. The group were able to round up the staff, holding them hostage as they prepared the breakout. However, Batman had learned their plans and arrived before they could do so. When the Children released the inmates, the vigilante worked to contain the inmates and pursued Lady Arkham through the building. The GCPD, having been tipped off by Batman, were also able to arrive before the inmates escaped, helping round them up. Due to these measures, the Children of Arkham's plans were unsuccessful and the rioting inmates were quickly rounded up.
During a press conference supporting the recently inaugurated Commissioner Gordon, Bruce Wayne can attend. During it, he can announce plans to cease construction on the Mental Hospital and instead focus on refurbishment of Arkham Asylum.
Following the riots, Arkham Asylum was given new counter measures to reduce successful escape attempts. Several months later, former Arkham staff active during Thomas Wayne's time were targeted by Deadshot and Black Spider. Deciding to use patient information to discover the former's identity, Batman broke into Arkham Asylum, using a variety of tools to bypass the security measures. In the record room, the vigilante was ambushed by Black Spider, who had also broken in to destroy these. Batman defeated the mercenary, leaving him incapacitated in the room, and used records to deduce Deadshot's identity; Floyd Lawton.
Deadshot eventually captured many of his targets, holding them hostage within one of his privately owned buildings. Batman tracked him down, freed the hostages and fought the criminal. After his defeat, Lawton was committed to Arkham Asylum, due to "untreated PTSD". However, he was later freed in return for being part of a suicidal job.
If Bruce announced plans to refurbish Arkham Asylum, he is told about its progress by Regina Zellerbach. Whilst undercover in the Pact, he can also find multiple items from the hospital, including Harleen Quinzel's overalls and photos taken by John. After John's transformation into the Joker and his subsequent arrest, he was returned to the Asylum. If he acted as a vigilante, he can be visited by Bruce (if regarded as a friend) or checked by Batman to ensure he hasn't escaped (if he didn't and gave up Alfred).