What the achievement asks
Buy every research item in the Titanic Research Archive. Ten points, account-wide, both factions.
The Archive is the talent tree that makes Horrific Visions survivable, bought with Corrupted Mementos earned inside the visions themselves.
The number is 69,600
That is the total cost of the full Archive, and it is the only figure that matters for planning. Mementos come from chests, kills and objectives inside each vision, so the Archive is paid for by the same runs that raise your cloak.
A talent with three ranks must be filled to 3 of 3. Putting one point in each and calling the tree complete is the mistake people make, and the achievement will not fire until every rank of every talent is bought.
It is the prerequisite for the hard achievements, not a reward
The order to do things in is worth being deliberate about.
A full Archive is what makes a five-mask vision possible, so this is not something to finish last. Two talents in particular are what people describe as essential before attempting the hardest runs: the one that revives you when you would otherwise die, and the one that gives a large burst of power.
That means the sensible sequence is: run visions for mementos, fill the Archive, and only then attempt The Most Horrific Vision of Stormwind or of Orgrimmar.
It pays for itself along the way
The same runs raise the cloak for Unwavering Resolve, and each rank of the cloak makes the next run easier, which makes mementos come faster. Nothing here is farmed separately.