The Elder Scrolls V: Skyrim Special Edition

BLACK BOOKS INTRODUCTION
There are 7 Black Books located across Solstheim, with 2 of them being acquired during the Main Questline.

You can choose any 5 books to pursue, and can find them at any time or in any order (with the exception of the 2 in the main questline).

Opening a Black Book transports the reader into a region in the realm of Apocrypha. Once the region has been traversed, a book awaits on a pedestal. Reading the book grants a unique power or ability, it is only after reading the book on the Pedestal you will get credit towards the achievement.


MAIN QUESTS
Black Book: Waking Dreams

Found in the Temple of Miraak during the main questline.

Note: While you acquire the book early in the Temple of Miraak, you will only learn it's secrets at the end of the main quest "At the Summit of Apocrypha".

Black Book: Epistolary Acumen

This Black Book is found at the end of the quest "The Path of Knowledge". This quest leads directly to the next quest "The Gardener of Men" where you must learn the secrets of this book.


FINDING ADDITIONAL BLACK BOOKS
After completing the main quest "The Gardener of Men", it is possible to ask Neloth for help discovering the remaining books, speaking to him (he lives in Tel Mithryn) will start the quest "Lost Knowledge", this quest is repeatable and will guide you to finding a random undiscovered book.

Alternatively, you can go directly to the Black Book without the help of Neloth and learn the secrets of all the remaining Black Books yourself. For the achievement you only need 5, so after completing the main quest you will only need to find 3 more.


BLACK BOOK LOCATIONS
Black Book: Untold Legends

After the word wall in Benkongerike. The passageway is at the top of a ramp formed by a fallen pillar to the left if facing the word wall.


Black Book: The Winds of Change

Bloodskal Barrow, accessed through Raven Rock Mine, after defeating Zahkriisos the Dragon Priest during the quest "The Final Descent".

This quest can be started by talking to Crescius Caerellius who will be encountered shortly after entering the Mine. The entrance to the mine can be found by heading North from the dock in Raven Rock.


Black Book: The Sallow Regent

In White Ridge Barrow. Reach the end of the dungeon and defeat Dukaan the Dragon Priest. This Book will be here along with a Word Wall for the new “Cyclone” Shout.


Black Book: The Hidden Twilight

Found in the locked room in Neloth’s house, which is located in Tel Mithryn.

After completing “The Path of Knowledge”, speak to Neloth and complete one side quest for him. After completing that, speak to him again to begin the quest "Reluctant Steward". Once you complete this quest you can unlock the door to his Staff Enchanter, in this room is the Black Book.

(When he asks you to find a new steward speak to Drovas Relvi at The Retching Netch in Raven Rock.)


Black Book: Filament and Filigree

The book can be found in Kolbjorn Barrow while completing the "Unearthed" quest (which is started after entering the Barrow). After defeating Ahzidal and confronting Ralis Sedarys, a passage to the southwest opens. Through that passage, the book is on a pedestal in the next chamber.

Note: This sidequest is quite long, so I don't recommend acquiring this book solely for the achievement.


^{You can use elderscrolls.wikia.com to find the locations of any places I have mentioned (it wouldn't let me embed their maps in the actual solution unfortunately)}


RECOMMENDED BOOKS
If you would like to be efficient with this achievement I recommend collecting the following books:

Black Book: Waking Dreams - Found during the Main Quest

Black Book: Epistolary Acumen - Found during the Main Quest

Black Book: Untold Legends - Found in Benkongerike

Black Book: The Hidden Twilight - Completing a couple easy tasks for Neloth. You also gain access to a Staff Enchanter.

Black Book: The Winds of Change - Found during the quest "The Final Descent", this quest is required to unlock 2 other achievements.

There are a total of 7 Black Books. Reading each one will take you into the realm of Apocrypha. Make your way through the various twisty paths and puzzles you encounter here, and you'll eventually reach another book at the end of the area. Read this book to gain a special power and return to Solstheim. (Some of the books will give you 3 options for a power to choose, so make sure to look at the green orbs hovering above the book for details on each one!)

Two of the Black Books will be encountered during Dragonborn's main story, so you'll need to track down at least 3 others. Most of the other 5 are linked to sidequests, but here are the locations of them all:

Waking Dreams - Main story (You pick up this book early on, but it doesn't actually count towards this achievement until you've finished the At the Summit of Apocrypha quest.)
Epistolary Acumen - Main story
Filament and Filigree - Kolbjorn Barrow (end of Unearthed sidequest)
The Hidden Twilight - Tel Mithryn (After finishing several sidequests for Neloth, he'll give you a key to the room containing the book.)
The Sallow Regent - White Ridge Barrow
The Winds of Change - Raven Rock Mine (You'll pick this one up since it's in the same area as one of the words for the Dragon Aspect shout and achievement.)
Untold Legends - Benkongerike

Credit to: alanp9