You may be cutting this achievement close toward the end of the game, so I would suggest putting quite a few points into your Speech skill early so you can work on it the entire time. While speaking with characters, you may be presented with dialogue choices that say "[Speech] %" with a number being in there for your success chance. The higher the speech skill, the better the chance you'll get away with whatever you're going to say. There are quite a few in the Underworld (at least five just after accepting the quest from Crowley, speak with other ghouls in the city), as well as at least one or two per quest. Speak with everyone possible as even some non-quest related people can have challenges.
There is an infinite speech challenge in Big Town to exploit. When you arrive, speak with Bittercup and agree with her that no one understands you. Once she talks about her dating Pappy, end the conversation and go talk to Pappy. Begin with "You came here with Bittercup, right?" and if he doesn't get upset you will win the challenge. Be warned though, if you do lose this challenge the loop will disappear. Try to make sure you have 100% success chance before using it.
I was able to find another infinite loop in Little Lamplight. You need high Intelligence for it though. Talk with Knock Knock and ask her what her job is. She will explain her role in helping morale and offer to tell a joke. Agree to hear it, but then you will have the option of interrupting the joke with an "[Intelligence]" dialogue choice. She will think your joke is funnier but tell you what hers would have been. You now have an option to lie and say her joke would have been funny too. Once you win that speech challenge, you will still be able to ask her about her job and repeat the entire dialogue over and over.