The conversations that the achievement refers to are the optional conversations you can have with people during the course of the game by approaching someone at your camp and pushing X when the prompt appears.
This achievement can be missed; you'll have to play through the game again if you miss any conversations. This also must be done in one play-through. Thankfully, there are only a few places that have conversations, and all of them take place near unmissable camps with people who are obviously there with you as part of the story. There aren't any hidden people to find.
The TR save icon will appear in the bottom right when you've successfully registered as having completed a full conversation with someone. Look for this to confirm when your game has given you credit for that conversation.
Some people say that the game will not always retain credit for having a conversation if you die before you reach a new checkpoint. If this ever happens, just go talk to the person again after your death.
Here are the conversations, in order:
1. In the Coastal Forest there are 4 conversations with Whitman before you upgrade your weapon for the first time, but people have reported that you don't need to get them to complete the achievement. The TR save icon does not appear when you finish speaking to him, either. All conversations after this are definitely mandatory, though.
2. In the Mountain Village, after Roth gives you the climbing axe, talk to him 4 times.
3. After the rescue plane crashes, Roth and Lara argue about whether she should go look for survivors. Talk to Roth 4 times before you set out.
4. Much later in the game, you meet up with some of your crew on Shipwreck Beach to repair a boat. After a cut scene, they will continue to talk to each other for a while. Wait until all of this is done, and then you can speak to Sam 3 times, Jonah 3 times, and Reyes 3 times. This must be done before you complete the next quest objective.
5. Very soon after #4, also on Shipwreck Beach. When you return with the part for Jonah speak to him 4 Times, Sam 3 times, Whitman 2 times, and Reyes once. Reyes will automatically start talking to you when you approach her, so don't count that as your one time. This all needs to be done before you meet up with Alex.
6. When you return to the camp on Shipwreck Beach after the event on the Endurance with Alex, speak to Jonah 3 times, Reyes 3 Times, and Whitman 3 Times. Doing all of this should immediately trigger the achievement.
If you don't get the achievement right after the final conversations, you have missed something, or something did not register; you'll have to play the game through again. Sorry.