All right, here's the deal. Junktown, it ain't what it used to be. The only reason it's still here is because of my gambling establishment. Even so, Killian's been breathing down my neck, and I'm starting to take it personal. And that's where you come in, slick.
While doing the Get Gizmo's confession quest, the Vault Dweller will have the option to record Gizmo hiring them to kill Killian Darkwater for him. Gizmo will ask to the Vault Dweller to return with the dog tags Killian wears around his neck as proof of his death. If the Vault Dweller does kill his mark, it'll anger all the guards of Junktown, who will attack them on sight thereafter.
The Vault Dweller will also get the reward for this quest if Kenji successfully assassinates Killian. The Vault Dweller can help Kenji without getting involved in the fight themselves, e.g. by stealing Killian's weapon and stimpaks just before Kenji initiates combat, or by standing in the door so that the guards standing outside won't be able to enter.
No matter the method chosen, retrieving the dog tags from Killian's corpse and bringing them to Gizmo completes the quest.