This achievement is a little complicated, but totally doable - especially at lower difficulties.
Create a new empire with the "Memorialist" Civic. This will allow you to create memorial structures. "Galactic Memorials" are the 3rd tier of the "Sanctuary of Repose". They require the planet to have 25 population and the "Planetary Capital" upgrade.
During the game, you have to create Tomb Worlds. There are a few ways this can happen. Primitive civilizations will occasionally kill themselves and create a tomb world, but this is random. As far as I know, there isn't a way for a player to change their own planet into a tomb world without getting the "Horizon Signal" (very unlikely) quest chain. To create tomb worlds, a player needs to destroy enemy worlds by:
- Bombarding an enemy planet with the "Javorian Pox" bombardment stance, granted from the Javorian Pox Sample Relic. Hard to get because you need the right precursor quest chain and the actually do the quests.
OR
- Bombarding an enemy planet with the "Armaggedon" bombardment stance. To get this you need to choose the "Become the Crisis" Ascension Perk and reach tier 3 in the Crisis Tab (next to the Relics tab). That tab will give you a list of options to build Crisis and get to tier 3, but wars with your enemies and espionage will get you there pretty quickly. The stance won't show up in your Policies tab, you have to select it by clicking on individual units and choosing the stance.
Once you unlock Tier 3, pick a war with an enemy weak enough that you can hold 5 of their planets long enough to totally bombard them (100% devastation and all of their population killed). It can take a long time if they're high population. Once all the population has been killed, the worlds will convert to tomb worlds. If this doesn't happen, check your bombardment stance.
Once you turn all five into tomb worlds, take those systems and end the war. You must have possession of those systems AND a way to get to them (otherwise you can't get colony ships to them and will need to have another way to get access).
Then colonize those worlds (DO NOT TERRAFORM) and build the Galactic Memorials on all five and you should get the achievement. You can move population from your earlier worlds to the tomb worlds to rush the monuments, and also move population from one to another once you build the Planetary Capital.