I was made aware of this secret achievement today, and apparently I've already missed some dialog between Lara and the Endurance crew members. During the course of the game you can actually talk to members of your crew when you meet up with them (I didn't know this and didn't ever try to). When do I have to talk to my crew members in order to hear the extra dialogs and get this achievement?
I, too, hadn't heard of this achievement until just now. Luckily, PowerPyx over at Xbox360Achievements had. He even put together a nice list!
He's also detailed the entire thing on YouTube.
One thing worth mentioning is that, if you have 50 salvage before speaking to Whitman the first time, it should give you all of the conversation options before you upgrade your axe, meaning you would be finished after clearing all of the dialogue with him (the chat option stops appearing).
Unfortunately, the achievement is missable, meaning you won't be able to get it without playing through the game again (assuming you've already missed one or two, that is).