(Optional) After completing Kings' Gambit, speak with Colonel Moore at Hoover Dam. She will ask you to assassinate Papa Khan. You may accept the task, or reject it, and instead continue with a more diplomatic approach. (Completing this step first will give the added options of having the Great Khans join the New California Republic or killing Papa Khan for them.)
Go to Red Rock Canyon, enter the only non-tent building in the area called 'Great Khan Longhouse'
During the daytime you will see Regis, Papa Khan and Karl sitting and eating behind a long table. Talk to Papa Khan. Exhaust all chat options until a long sentence asking Papa Khan to abandon the alliance with Caesar's Legion appears. He will refuse regardless. Exhaust all chat options with Regis and Karl. Exit the Longhouse; Regis will speak to you about how to convince Papa Khan to ally with NCR instead of the Legion - obtaining the support of the 4 advisers (Regis, Jack, Diane and Melissa) & discrediting Karl. To obtain Regis' trust, you need to find him proof that the Legions make slaves of their allies. Stealing Karl's journal from his footlocker and using it to disgrace him in front of Papa Khan will gain you Regis' support at the same time (using Speech to disgrace Karl will leave you unable to use Karl's journal for Regis' support). Proof can also be found in Caesar's tent in his fort. It's a slave ledger that you will find on one of the tables in Caesar's tent. To obtain Jack & Diane's trust, you only need to convince one of them. They are located due North, through the canyon at the Red Rock Drug Lab Either pass a Speech check of 60 with Jack or find Anders for Diane. He is in Cottonwood Cove, crucified at the entrance of town. To obtain Melissa's trust, simply visit her at the Great Khan Encampment and tell her that women aren't allowed to serve in the Legion.