Amadeus Arkham was the only son of mentally-ill Elizabeth Arkham, Amadeus watched his mother struggle with her illness for years, which drove him to become a psychiatrist. Years later, having earned a residency at the Metropolis State Psychiatric Hospital, he returned home, and having found his mother to be growing worse, aided her in committing suicide. After her death, he became the sole owner of her home, the Mercey Mansion, which he decided to turn into a facility for the mentally ill. He gave up his practice at MSPH and moved his wife Constance and daughter Harriet into the mansion.
During the remodeling process, one of Arkham's former patients, Martin "Mad Dog" Hawkins, escaped from Metropolis Penitentiary and broke into the Arkham family living quarters. Amadeus returned home that night and discovered his wife and daughter had been murdered. It was determined his daughter had been raped and decapitated. The words "Mad Dog" were carved into her chest.
When the Asylum opened that November, Hawkins was captured and became one of its first inmates. Amadeus insisted on treating him personally. After six months of treatment, Amadeus could see no improvement in Hawkins and killed him with a lethal dosage of electroshock therapy. The incident was ruled accidental but was only the beginning of Amadeus' slide into insanity, and eventually, he ended up dying as an inmate in his own asylum.
In other media
Batman: Arkham Asylum
Because the character is deceased by the time of Batman: Arkham Asylum, he doesn't physically appear in the game but his character profile is unlockable and the player can find his grave in the Asylum's cemetery. The grave appears however, to be dug up, as Arkham's skeleton is partially visible.
His diary entries are recorded in a preview for the game, showing footage of the asylum and its patients:
Journal Entry #1, June 4th, 1920
"Mother is dead. Dementia took her. With my knowledge, I can begin working to cure the world's ailments. The name 'Amadeus Arkham' will become legendary."
Journal Entry #2, September, 1920
"A radical step must be taken to combat Gotham City's diseased."
Journal Entry #3, February, 1921
"Construction has begun on the Elizabeth Arkham Asylum for the criminally insane. Finally, I will be able to treat my patients properly."
Journal Entry #4, April 1st, 1921
"Returned home today to find my family murdered by my patient Martin Hawkins. I feel oddly detached."
Journal Entry #5, September 17th, 1921
"Today, I begin treatment of Martin Hawkins. I will rehabilitate this man."
Journal Entry #6, April 2nd, 1922
"During treatment with Hawkins, I resorted to...extreme measures. It proved more than he could sustain."
Journal Entry #7, June 4th, 1923
"Gotham City is lost. The lunatics are irrepressible. Incurable. The only sensible treatment: Eradication."
Journal Entry #8, October 18th, 1923
"Am I a doctor, or a murderer? I can no longer differentiate. I will give my last breath to deal with the filth that infects Gotham City!"
Batman: Arkham Origins
Days before attending his forty-fourth birthday party, city architect Cyrus Pinkney was approached by Henry Cobblepot, intending to collect a loan he owed. Though Cobblepot got the money, Pinkney betrayed him and asked the mayor to deny Cobblepot building a munitions factory which would require tearing down part of the original Gotham. Eventually, Pinkney met a young university student named Amadeus Arkham, who explained to him his plan for a hospital for the mentally ill. Pinkney, pleased with the young man's plan, agreed to help him with building the hospital that would become the hell hole that is Arkham Asylum. The two of them became fast friends, and Pinkney soon let Arkham in on Henry Cobblepot's plan to poison him on his forty-fourth birthday party. Arkham formulated a plan that would not only save Pinkney's life but would also end Cobblepot's. He swapped the poisoned wine for one spiked with a drug that would put him in a death-like state, which would fool even professional physicians. The plan was a success and Cyrus Pinkney, now labeled dead, got his revenge on Henry Cobblepot by severing the brake line of his car, causing his death via a car crash.