I'm on For the Republic, Part 2 quest, and I'm suppose to go talk to Papa Khan to break the Alliance. The only option I had at Red Rock was to talk to Regis. I go back to Colonel Moore and tell her about how Regis would make an alliance if he was in charge. At the end of her conversation, I had a choice of saying to agree to kill Papa Khan or try a diplomatic solution. I chose diplomacy because it is not dead. This starts Oh My Papa and then I travel back to Red Rock and I get no more dialogue options from anybody to do anything about the Caesar's Legion alliance.

I couldn't figure it out so...I killed Papa Khan. I fail Oh My Papa and then I travel back to Colonel Moore. She has no new dialogue options and I can't tell her I killed Papa Khan. What do I do?

I'm trying to side with NCR and also trying to get the hardcoe Mode achievement. I'm not sure if doing anything in the console will prevent me from getting the achievement. If so, I'd like to avoid that.

From what I understand, it can be difficult to do this peaceably. I've never done it that way so I can't give personal experience advice. Maybe others can. But the wiki link has a whole list of all the possible methods & issues: http://fallout.wikia.com/wiki/Oh_My_Papa

Using the console does stop achievements from progressing when you use them for that play session, but what you do is save game (in a new slot), exit game, and restart game. The change should still be there but the console is deactivated (so to speak) again and your achievements should now progress. Some of the console command effects don't seem to persist in a save game (like god mode, you have to reuse console/retype...) but I believe things like rep/stat changes and various other such things do.

You have to talk to Regis about ending their alliance with the Legion first.
He'll tell you you have to convince all Papa's lieutenants that the alliance is a bad idea (just follow the quest marks for Oh my papa).

When you've convinced all the lieutenants that they shouldn't be allied with the Legion talk to Papa to break the alliance.
After that you can talk to Regis and Moore about an Khans-NCR alliance, and you'd still have to kill Papa. But you have to break their alliance with the Legion first

It's been a while since I've done this quest, but I think you have to talk to Papa Khan and one of the guys sitting next to him.
After that, one of the Khans should meet you just outside - he should tell you how to talk to / convince of breaking the alliance
with Caesar's Legion. After this is done, talk to Papa Khan again (you don't have to kill him) - he will ask for an inspiration for
the Khans. I think you'll be sent to the Followers and the 188 post to obtain a book. Anyway, good luck!