This is actually one of the more enjoyable boss fights find in the game. It's very reminiscent of the Mr. Freeze battle from Arkham Ciy. To find Deadshot, head to the Merchant's Bank and enter through the roof entrance after you have solved the helicopter crash investigation.
When you enter the bank, there will be numerous gunmen wandering about along with Deadshot himself who will choose to stay along the upper walkway. To take down Deadshot, you'll want to take out some of his gunmen first and then focus on him.
To hurt Deadshot at first, stick to the gargoyles at the side of the room and then quietly drop down behind him and sneak up on him. Perform a silent takedown and then get out of there as fast as you can by grappling up to another gargoyle.
Once you have taken Deadshot down a couple times, some more enemies will pour into the bank and Deadshot will now prefer to stick to the bottom level of the bank. Use the grates and vents around the lower floor to get a few more silent takedowns on Deadshot and he will then go and take a hostage in the middle of the bank.
If Deadshot sees you once, he will shoot the hostage and you will fail the mission. It can be a little tricky, but the best way seems to be to hangout on the gargoyle where you first entered and watch Deadshot turn around by seeing where his laser is pointed.
When it is away from you, drop down to the floor behind him and then sneak up and take him down to end the standoff once and for all.
For more tactics here's our full Batman: Arkham Origins boss guide!
