In total there are 9 playable rulers (kings and queens) to unlock, their requirements are not too difficult and you will be able to unlock each of these characters by normal game progression anyway. You will find it more beneficial instead to focus on meeting objectives of other aspects of the game such as colllecting all the deaths and seeing all the cards. If however you are intent on unlocking these characters quickly the guide below will help you.
Daenerys:
Queen Daenerys Targaryen, also known as Dany and Daenerys Stormborn is the default playable character. She is the first character you can choose when you start the game.
Tyrion Lannister:
Lord Tyrion Lannister is the youngest child of Lord Tywin Lannister and younger brother of Cersei and Jaime Lannister. The first time Daenerys is killed you will automatically unlock this dwarf at the character selection screen.
Cersei Lannister
While playing as Tyrion one of the goals will be to oversee the trial of Cersei. When Lord Varys tells you Cersei has been found you need to capture her and attend the trial. When you do this you will gain access to the card The Mad Queen, which allows you to play as Cersei.
Jaime Lannister
To unlock the younger twin brother of Cersei, and older brother of Tyrion Lannister you must play as Cersei and survive long enough for the White Walker threat to become real. At this point if you allow Jaime to mine Dragonstone he will become unlocked as a playable character.
Sansa Stark
Sansa is the eldest daughter and second child of Lord Eddard Stark and Lady Catelyn. She will approach you and request to buy Wildfire with the intention to then blow up The Twins, the castle of House Frey. As long as the ruler (whoever you have chosen) agrees to Sansa's request, you will then witness an 'unexpected use of Wildfire', and Sansa will become unlocked for future playthroughs.
Jon Snow:
Born Aegon Targaryen, Jon Snow is the son of Lyanna Stark and Rhaegar Targaryen, the late Prince of Dragonstone. The former Lord Commander of the Night's Watch becomes accessible as a playable character when you play as Daenerys and he appears and talks to you. Jon Snow then becomes unlocked when Daenerys next dies.
Arya Stark
Arya Stark is the third child and second daughter of Lord Eddard Stark and his wife, Lady Catelyn Stark. Arya is unlocked while playing as Tyrion. Once advised that an ally of Cersei has been killed, you will be able to investigate to try and find the culprit who turns out to be Arya Stark who becomes unlocked. Selecting Ayra as a playable character will then lead to her assassinating Cersei and taking her place on the throne.
Gendry:
Gendry is a skilled blacksmith and an unacknowledged ******* son of King Robert Baratheon. To access Gendry you must play as Arya. The blacksmith will approach the queen regarding an excess of iron, and should be told to make swords. After the blacksmith is called 'filth' by the Master of Coin, disagreeing will allow Arya to expose the blacksmith's true identity unlocking him as a playable character.
The Three-Eyed Raven:
The Three-Eyed Raven was the last Greenseer, a human living among the last of the Children of the Forest beyond the Wall. He becomes accessable when you have have unlocked and played as all of the other characters.