***There is a "work-around" to still use this solution. As of the 10th of Dec 2015, a patch was put in place to correct the glitch that allows this solution to work. If you have downloaded the latest update of the game, you will need to take a few extra steps to get the original solution to work. This is a method to exploit the bug in the original version of the game that let you pop this achievement by beating the game on a lower difficulty and then playing the last level of the game on Survivor. This worked 100% of the time before the patch, but if you follow this solution, please be aware that based off the rating of this solution, the success rate may be 75-90% now. (It could also be that I get some negative votes because some gamers may think it is unethical to pop an achievement without fulfilling all the requirements or can't follow instructions.)
Lets start with mentioning that you must have a physical copy of the disc to get this to work now. Simply put, once you beat the game on a lower difficulty, you can uninstall your game, go offline, re-install, and follow the original solution.
It is important that you keep your original save file intact. Making a copy of the save could be helpful. "AHuge RedDragon" helped by laying out the steps to keep your save file on Xbox One which can be more confusing then it was on the 360:
"Open games & apps --> Highlight Tomb Raider --> Menu button --> Manage game --> Select the main game tile --> Menu --> uninstall. DO NOT press uninstall all like you usually would when finished with a game"
One last thing to mention, the players that have confirmed that this method can still work have reported that since playing offline, you lose the ability to use cards to make things easier. (To be clear for the "work-around": If Chapter Elite will not work due to an error, any type of Chapter Select will also get you the achievement). Follow the original guide, go back online, download the update, and the achievement may not always pop right away. If it does not pop, it should show as complete. ***
There is currently an easier way of grabbing this achievement, if you have completed the game on any other difficulty mode. This method definitely needs it's own solution.
-Go to Expeditions in the Main Menu
-Set Mode to: Chapter Replay Elite (This is like New Game+ where you keep all your skills and upgrades)
-Set Level to: Chamber of Souls
-Set Difficulty to: Survivor
All you need to do is beat the last section of the game, but on Survivor difficulty.
This mode also gives you the opportunity to use up to 5 unlocked cards to make the level easier. I highly recommend using unlimited poison arrows or unlimited explosion arrows if you have those cards unlocked. Even if you don't, load up all 5 card slots with ones that will make things easier for you like: big head enemies, a chance that an enemy explodes in a poison cloud on death, take less damage, see enemies through walls, etc.
I can confirm that this works. I used this method, and there are many other reports on this site and others that mention doing this.
After completion, the achievement will pop at the end of the cinematic. I am not sure if this will get patched at some point, so I would go for it as soon as you complete the story on an easier difficulty.
*If this solution helped you in some way, please vote positively. If you think it needs improvement, please let me know how in a private message or by leaving a comment.*