Credit to Godfather A7X for the bulk of the text, I just tweaked the specifics.
To get an invitation to the Sanguine Pact Secret Society you must clear Barbarian Camps, each one has around a 80% chance to generate an invitation with odds decreasing as more civ's join said society.
Promotions within Secret Societies unlock when entering a new era with the fourth and final being unlocked in the Atomic era, you will also need 4 free Governor points to assign and promote the Voidsingers Society.
Keep in mind you can only join 1 Secret Society so don't join others should you find them before the Sanguine Pact.
Set up a custom game as follows:
Map: Any size, standard is good enough and on Continents
Put city States to 0
Turn off all victory conditions except Domination
Starting Era: Information
Turn ON Barbarians
Make sure to turn on Secret Societies Mode under the Game Modes Tab
Pick any Civ to play as and remove all others except for 1
Now settle 1 city right, then spread out to find a barbarian camp; I turned my Spec Ops in an Army, so they should have no issues killing the single AT Crew at the Camp. You should find The Sanguine Pact pretty quickly, then Assign them to your city and promote 3 times to the max level (you will have 10 Governor points available) and the Achievement is yours.
This can also be done in Hot Seat if you prefer, though it takes around the same amount of time regardless.