Deadshot is the main antagonist in Batman: Sins of the Father; a series of tie-in comics set between the first and second seasons of Batman: The Telltale Series. A gifted shooter and assassin, he targeted the former staff members of Arkham Asylum in retribution for a traumatic event in his life. Deadshot's true identity was Floyd Lawton, the son of two of Thomas Wayne's former allies.
The son of real-estate tycoons George and Genevieve Lawton, Floyd was the victim of abuse, both physical and psychological, at the hands of his parents. However, he was close to his brother Edward, who idealized fighting for justice and heroic figures like Robin Hood and Zorro. Eventually, the Lawton brothers called the authorities and reported their parents, leading to their incarceration at Arkham Asylum. For three months, the two would live away from their parents in the care of a foster home.
However, the Lawtons were eventually released to return to their sons, due to a deal they had arranged with Dr. Thomas Wayne. Though they spent months hiding their true nature from the authorities and social workers, they eventually reverted back to their ways once they knew they were not under surveillance. After nine months, Edward was attacked whilst trying to protect Floyd from his parents, resulting in him being knocked down the stairs and breaking his neck. Scared that he may fall a similar fate, Floyd was forced to lie about the circumstances of Edward's death, placing him further into his parents' care.
After some time, Floyd suggested that he was to join a boarding school, hoping to get further away from his parents. Seeing it as an opportunity to improve their social status, the Lawtons agreed and enrolled him in one. Whilst their, Floyd excelled in both academics and sports, including tennis, riding and shooting. Excelling in the latter, his talents won him many award and competitions, something which only increased his parents' bragging. At the age of 14, Floyd murdered his parents, staging it as a murder-suicide at his mother's hands to avoid suspicion. Upon reaching the appropriate age, Floyd inherited their company and, seeking to keep his skills refined, turned it into a private security company. Through it, he became a skilled mercenary and refined his skills with using most forms of firearms.
Around the time the Waynes ties to both Carmine Falcone and Hamilton Hill, along with Thomas' illegal use of Arkham Asylum, Floyd realized that Wayne had been responsible for letting his parents back into his life and, indirectly, Edward's murder. With the billionaire now dead, Lawton decided to target staff at Arkham Asylum who were linked with Wayne's activities and assassinate them. To that end, he would hire professional mercenary Johnny LaMonica/the Black Spider and began hunting down staff involved in Thomas Wayne's activities.
When the families of Thomas Wayne's victims decided to sue Bruce Wayne and Wayne Industries for compensation, Bruce gathered them all to Wayne Tower to discuss terms.
Suddenly Deadshot open-fired on the building and succeeded in killing a man, causing the rest of the group to take cover. Bruce was forced to leave the group to confront Deadshot as Batman, but the villain escaped before he could get to him.
Later, Bruce discovered that the killed man and many people among the group he invited were members of Arkham Asylum staff. Leading him to believe either they could've been the intended targets, or Bruce himself.
With no leads as to Deadshot's whereabouts, Batman scans the city and locates Black Spider, another assassin who might be working for the same person who hired Deadshot.
However, while Batman fought him, Black Spider revealed that he was actually hired by Deadshot, who then shot Batman in the chest. Luckily, the Batsuit protected Bruce from the bullet and he proceeded to fight both villains until they managed to escape.
In order to track Deadshot's movements, Batman breaks into Arkham Asylum to find something on the gunman's true identity. Despite a brief tussle with Black Spider in the records room, Batman got what he came for and returned to the Batcave to learn more on Deadshot's past.
Bruce discovered that Deadshot was a member of the Lawtons, a rather wealthy family in Gotham with an unfortunate history. George and Genevieve Lawton were accused of violent abuse towards their two sons Edward and Floyd Lawton. They were arrested and sent to Arkham Asylum to be evaluated by Dr. Thomas Wayne, who was willing to deem them model parents in exchange for a building that later became Wayne Tower. The pair were soon released after this and still allowed custody of their children.
One year later, Edward Lawton was found dead by his brother Floyd at the foot of the stairs. The official statement was that the boy fell down the stairs and broke his neck. Although Bruce implies that his parents were to blame. This event caused Floyd to avenge his brother by murdering his parents. After he obtained a gun, Floyd shot his father from across the room before he ambushed his mother and shot her in the head; leaving the gun in her hand to make it look like suicide.
Floyd inherited his family's fortune, eventually using it to start his own business in private security and military contracting. He also frequently visits chaotic war-torn countries where he could keep his skills sharp without any repercussions.
Bruce believed that Floyd continues to kill people simply because he enjoys it, and since he wasn't intimidated by Batman, confronting him as Thomas Wayne's son might spark a reaction instead.