Floyd Lawton is Deadshot, the assassin that never misses.
Before Brainiac's Invasion
Though Deadshot remained completely absent from the five years of Superman's regime following the Joker's attack on Metropolis, the deadly assassin appeared after Superman's fall as a member of Amanda Waller's Suicide Squad.  However, Lawton was quickly liberated from his work with the government when a mysterious Batman impostor commandeered the squad for his employer, forcing Deadshot and the other members to work for him. 
When Brainiac arrived on Earth to conquer and study it with his forces, Deadshot was working with Gorilla Grodd's Society determined for world domination. Deadshot was tasked with Captain Cold to take out the Flash if the hero were to come out of retirement.
As predicted by Brainiac, the speedster did, in fact, come back, forcing a fight between the fastest man alive and the man who never misses, at which point Deadshot revealed that he was not voluntarily working for the Society but being forced to by Grodd, who had the remote detonator to the bomb in Deadshot's head that was planted there during his time with the Suicide Squad.
Defeated by the Flash, Deadshot and the other Society members later discovered that Grodd's goal was not actually world domination and, instead, he was working directly with Brainiac. Deadshot quit the team, despite the bomb in his head, and was eventually totally freed when Grodd was killed by Aquaman. 
Possible Post-Brainiac Epilogue
This section of history takes place in a timeline outside that of normal Injustice: Gods Among Us history. Information here does not reflect the true history of neither the Injustice: Gods Among Us universe nor the Injustice: The Regime universe, and should not be seen as relevant to future sequels or comic adaptations.
"Grodd used the damned nano-explosive in my head to make me his bitch. The jobs he made me do ... well, let's just say they crossed some lines. I bet taking Brainiac and Grodd into the authorities would beat doing another stretch in Belle Reve, ... and I was right. Bruce Wayne even bankrolled getting rid of my TNT migraines for good. I frickin' hate hospitals, but god was it worth it. With that nano safely out, I could finally see Zoe again.
Last time I saw my little girl, she had nine candles on her birthday cake. Now I gotta second shot at being Dad - and I sure as hell don't plan to miss it.
- Firearms: Deadshot is the best marksman in the world. He's known to never miss.
- Hand-to-Hand Combat (Basic)
- Weaponry: Deadshot is highly skilled with both guns and knives.
- 19 Appearances of Floyd Lawton (Injustice)
- 7 Images featuring Floyd Lawton (Injustice)
- Quotations by or about Floyd Lawton (Injustice)
- Character Gallery: Floyd Lawton (Injustice)
Suicide Squad member
Secret Society of Super-Villains member