You must go through all converstaions with all of the main characters (Deckard, Adria, Leah and Tyrael), followers (Templar, Enchantress, Scoundrel) and artisans (Blacksmith and Jeweler). Some of the conversations are very time-specific, being available during certain points of the main story, but thankfully you can always load up an older chapter, or just catch the conversation on your next playthrough.
To find your conversation progress while in-game, hit select challenges, and highlight General. The challenge related to each character is listed on the quest sheet below in parentheses next to the character.
Notes: For the follower chats, you will need to unlock the followers with each character before you can talk to them on that character. Progress carries over between all of your characters, and unlocking the Jeweler and Blacksmith on one character will allow you to find them from the start on another character.
Note that some of the conversations might not update on the in-game challenge list, but will still work for the achievement. Shen's King Leoric conversation did check for me after repeating the conversation a few times, but completing the rest of the list gave me the check mark and the achievement.
A full conversation list can be found here