To get this trophy you must complete the quest "Elder Knowledge" which is a part of the main story and cannot be missed.
*** Spoiler - click to reveal *** After speaking with Paarthurnax from the "Throat of the World" quest, he will task you with finding an elder scroll so you can go back in time and learn the Dragonrend shout. Speak with either Arngeir or Esbern about the scroll and they should direct you to the mages in Winterhold. If you have not discovered Winterhold yet, you can use the carriage for 50 gold.
Walk towards the entrance of the College of Winterhold. If this is your first time visiting the college, you will have to will have to speak with Faralda. To get past her, you will either have to pass a persuasion check, demonstrate your knowledge by showing her some spells (check the "Gatekeeper" trophy guide. Or showing her that you are Dragonborn by shouting. One you have done one of these choices, you should gain access to the college was well as unlock the "Gatekeeper" trophy.
Go into the Hall of the Elements and take a right and open the door to ehe Arcanaeum. Speak with Urag bro-Shub and ask about the Elder Scroll. He will return with books about Elder Scrolls. After you open them up you will begin the quest "Discerning the Transmundane". Which you will have to do in order to learn the location of the Elder Scroll. The first task in this quest is to speak with Urag about the books he just gave you.
After speaking with Urag he will tell you to go find Septimus Signus. Leave the Arcanaeum and head towards your quest marker up north. After a short amount of time you should arrive at Septimus Signus's Outpost. After speaking with Septimus, he will inform you that the scroll is in Blackreach which is underneath a place called Alftand. At this point forward there is going to be a large dungeon with a lot of enemies and loot, so I recommend stocking up on potions and managing your inventory before going in.
While making your way through the dungeon you will encounter various enemies such as dwarves spiders and spheres. You will eventually reach a large open area where you have to use a lever to open a gate and progress. In this area you will have to fight a Dwarven Centurion so make sure to save before going in. After you kill the centurion, loot its body to get the Aftland Lift Key. Move forward up the stairs towards the dwarves mechanism. Activating it will reveal the entrance to Blackreach.
Blackreach is a large open area with lots of places to explore however your should not get lost if you follow your quest marker. I do recommend heading into Sinderion's Field Laboratory which is the building directly infant of you as you walk in. In there you will find lots of ingredients as well as alchemy and enchanting tables. This is a great place to create more potions. Make your away across Blackreach until you reach the elevator to the Tower of Mzark.
Once inside, go to the large Dwemer artifact in the middle of the tower and place the blank lexicon in the holder. Now you will have to push the buttons in the rolling order.
Middle-right - 4 times. The lexicon should start glowing
Middle-left - 2 times.
Far-left 0 once.
The artifact in the middle should open up, revealing the elder scroll. Grab the scroll, completing the quest, and receive your trophy.
*Cross-posted from Skyrim (PS3) trophy guide.