Kirito mode will be unlocked story mode wise, you will not miss it. Go to the main menu and press start game (don't load). You will get two options, your OC or Kirito mode, click the latter.
Death Gun will be the last boss in Kirito's mode. You will have to fight multiple goons before meeting him but it shouldn't take more than half an hour. I did have some trouble fighting him but as long as you have the self healing gadget you should be fine.
Kirito mode will be unavailable at some point if you decide to fight the final boss in the game. You will be warned so make sure to finish this first!