In order to unlock this achievement, you must complete Memory 4 without failing once. Failing before "entering the Animus" (falling down the hole in the Animus room) does NOT count against you. (credit to Aldanga for this information)
Now this is the hardest achievement in this DLC pack, it took me multiple tries but really it's just trial, error and knowing exactly what to do as you make it to the end. Before you know it you'll be playing through and mastering every section without a problem.
The only two sections I had major issues with, the first involved the yellow lasers that have to be blocked by one of your blocks in order to pass through. When you reach these lasers, the first one just set a block down (make sure your blocks are highlighted yellow or they won't block the beam) on the far left and pass by it, setting down the flat blocks to get your balance. You'll see a huge wall of lasers, what I did was I moved to the right and set two blocks side by side to give me enough walking room through the beams (as your blocks will constantly be moving left here, you'll want to move fast overtop of your walkway otherwise you'll get caught by the beams). These beams are blocked from the bottom of your blocks, keep that in mind.
The next beam to worry about will give you a small area of space to move to the next area. Start by bringing yourself over to the left and set down a block right against the beam and quickly move yourself to the right and create a path to get you through.
The second area that gave me a major issue is the very last sequence where you're falling and you have to create blocks and avoid hitting those huge purple shapes. It takes patience as you're constantly being pushed down so you won't have a whole lot of time to position yourself but it's still enough time to make it work. There's three types of shapes here: a wide range one with a cut in the structure allowing you to pass through (it spins constantly), two shapes that constantly move to opposite sides of the screen (giving you two openings at any given time, sort of like a checkerboard) and a set of shapes that take up half the screen and alternate. Everything before the "break zone" is fairly simple to navigate, however the last stretch has you faced with two wide area shapes at the start and a bunch of crap to worry about on the way down. Just patiently tackle this and it will come in time.
Credit to Hazey Fiasco: If you switch your block to the wedge shape (basically not the flat one) and set it down as you're falling (make sure you're a safe distance below the above obstacle), you'll be able to move up a bit giving you more time to adjust yourself for the next drop. It may be a good idea to alternate your blocks if you're worried about landing on a wedge block and walking up into an obstacle.
The achievement pops as soon as you land at the bottom of this section, facing the doorway. You don't have to enter the doorway for it to pop.
Edit: Adding a video of the path I took to getting this achievement, apologies for the quality.
Edit2: After doing it on the Ezio Collection, I added a new video for that run.