To get to War Stories, select your server, pass the initial loading screens, at the main menu select 'War Stories' (above 'Multiplayer')
To get this achievement you must complete every chapter of four different stories. There are currently eight stories. Five stories contain three chapters and three stories contain one chapter. Only the main objectives of the missions need to be completed, the secondary objectives are not necessary. Single parts of Spoils of War do count as individual stories. (Credit to iDeary 97 & Hanrec2 for the last sentence.)
1. Brothers in Armor
2. Flashpoint Berlin
3. Operation Sealion
4. Kennedy's War
5. Runaway Tiger
6. Sieben Ritter
7. The Connoisseur
8. Potemkin
To rotate from different Stories and to select chapters, at the War Stories garage screen hit the B button once for chapter select and twice for story select.
This achievement is not retro active. If you play the stories on Xbox 360, they will not count towards the achievement and you will have to play them again on Xbox One. (Credit to Darknes727420)
Along the way you should also unlock;