First off you need to get Karl out of the picture. When they are all eating at the table either go to Karl's room and steal a journal out of his footlocker and give it to Papa Khan or if you have high enough speech you can make Karl say stuff that gets him killed immediately after.

To convince the Jack and Diane you can talk to Diane about Caesar and his ban on them...but you need to find proof. I went and talked to Jack and was able to convince him that he would be killed and Diane would be sold (I'm not sure if this needed high speech or not.

To convince Melissa - a lot of people have trouble finding her - you have to go through a family of Deathclaws in the Quarry. Fast travel to Sloan or walk there if you don't have it and it leads you to the mouth of the quarrry. I haven't tried it but I'm sure you could always solo stealth boy your way through (For PC: And if you're REALLY having too much trouble and you're about to give up you could always open up the console and toggle God Mode with the command tgm).

Finally to get the proof for Regis you have to go to Cottonwood Cove -> The Fort -> Caesar's Tent and on the left there is a table with a slave ledger. This doesn't count as stealing. Then you just have to go and give it to Regis, and talk to Papa Khan.