38 years ago - Floyd Lawton is born, the son of a violent, abusive father.
26 years ago - 12-year-old Floyd tries to kill his father but misses, hitting his older brother.
21 years ago - 17-year-old Floyd joins the army to escape his father.
18 years ago - 20-year-old Floyd becomes an army sniper.
17 years ago - 21-year-old Floyd goes AWOL from the army to become a mercenary & assassin.
14 years ago - 24-year-old Floyd has an affair with Michelle Torres, unaware when she has his daughter.
12 years ago - 26-year-old Floyd competes for Carmine Falcone's 50 million dollar bounty on Batman. He kills Patricia Trayce to become the second best Merc on the planet, making himself a rival of Slade Wilson.
8 years ago - 30-year-old Floyd discovers that he has a daughter. He goes to war with a local crime lord to make her home safe. He ultimately gives up to Black Lightning in exchange for his protection of his daughter.
3 years ago - 35-year-old Floyd is denied his early release for killing a senator. He stays with the Suicide Squad's new membership.
2 years ago - 36-year-old Floyd uses the chaos of Atomic Skull's attack to help Catman escape the Suicide Squad.
Deadshot had a pretty ignominious beginning; he debuted as a top hat & tails wearing sharpshooter that was supposedly competing with Batman as a potential hero in Gotham before he was revealed to be a mercenary. It was the sort of one-off character that you never hear from again, before someone decided to re-purpose him as a vengeful assassin with an immediately striking costume. He's a very cool character that would absolutely be considered the most lethal killer-for-hire in the world of DC if that title didn't already belong to Deathstroke.
It might actually be the fact that he isn't the number one guy that gives so much leeway with the types of stories you can tell with him. He's such a mainstay of the Suicide Squad that he's probably the only character that HAS to be there, but he's also been depicted as both an utterly remorseless assassin and a tragic loner so lost in his own self-hatred that he is almost a tragic hero.
The innovation of his lost daughter and his own desire to do right by her came later on, but it's such a great story hook that we used it as the reason he's imprisoned and recruited by the Squad, a story that it seems will also be used in the upcoming Suicide Squad movie. We've also used his time in the Secret Six and his relationship with Catman as an element in Floyd's timeline, but ultimately he just has so many cool stories to his name it just becomes a matter of letting him be who he is.