In order for his business enterprises to flourish, Gizmo needs Killian Darkwater out of the way. The two have been locked in a stalemate for a while and the only way to resolve the deadlock is for one of them to fall. Gizmo's method is simple: Death.
Killing Killian can be done at any time while in Junktown, without having to go to Gizmo. The fat man will pay up regardless. However, going to Gizmo first and then going to Killian's will prevent Kenji from spawning. On the other hand, it allows the player to get a 1500 caps for taking Killian out instead of the usual 1000 (Intelligence 6 guarantees a success).
The easiest way is to just shoot Killian in the face until he dies, but this will also cause all Junktown guards and Lars to become permanently hostile.
Alternatively, with a well-developed unarmed or melee build, the player can enter Sneak mode and try to kill him in the first turn. Using firearms won't work, as they are loud and the guards aggro immediately.
An expensive, but reliable way is to use eight stimpaks on him in quick succession, then rest for ten minutes. The shock from the healing chems will kill him. You need eight, as each deals 9 points of damage after ten minutes; Killian has 64 hit points, so eight will deal 72 damage, while seven "only" 63.
Finally, the meta-assassination involves setting Killian up so that Kenji kills him. Before talking to Killian, the player should steal at least Killian's stimpaks (taking the Desert Eagle greatly increases Kenji's lifespan) and then talk to him. After exiting conversation, Kenji will spawn outside the shop and start walking in. The player then needs to stand in the doorway to prevent guards outside from intervening. It might take a reload or two, to ensure Kenji gets a good critical hit or two, but Killian will die and the remaining town guard will finish Kenji off.
Regardless of method used, the player needs to loot the dog tags from Killian's corpse and deliver them to the fat man. Gizmo will be jubilant and tell Izo to pay up. Talking to Izo or letting him interact with the player character is necessary to get the money, as it's not automatically deposited in inventory.