For the achievement you need to craft all 8 themes and for each theme you need to collect 9 different fragments. You can find these fragments in the wasteland on quests or just exploring.
A way you can make this achievement easier instead of just the normal grinding is by using your nuka cola quantums to get you the remaining missing pieces. They cost 3 quantums per fragment. This means one whole theme is 27 quantums and 216 total quantums if you bought them all with quantums. You can get duplicates, so just grinding this one makes it pretty frustrating.
An easy way to get nuka colas if you just downloaded the game is start the quest "Bottlecap Blues" which is available for a limited time, and should still be available. At the end of this five part quest you get characters called Bottle and Cappy who come into your vault every now and then and dance around you fault. When cappy dances, and you click on him,he gives you caps, and when bottle dances he gives you caps about half of the time and the other half he gives quantums (around 1-3). They come in fairly often if you just leave the game open so it makes the quantums a lot easier to come by.
To cut down on the actual crafting time it is important to use the correct S.P.E.C.I.A.L. stat for each theme. Railroad themes themes need Perception, Institute themes need Intelligence, BOS themes Strength, Minutemen themes need Agility. You should get it down to around 2 days if you use your dwellers correctly (or as low as 18.5 hours if every dweller is correctly maxed as Skelebone stated). Thanks SwaggaGround for the info!
BUG FIX
Some people are having a bug where they have all the correct materials (fragments, junk, caps) but they still can't select to craft it. It seems all you have to do is back out to the main menu and reload your vault and you should be able to craft then!
Thanks Arador for the solution!!
Good luck hunting!