On Tuesday, the online streaming website unveiled a new extended trailer for the show, which sees the stars' characters face up to the challenges of a being young royal couple.
The three-minute long clip for the 10-part series begins with a 25-year-old Princess Elizabeth (played by Claire) learning that she is now the new Queen of England, while Prince Philip (Matt) tries to adjust to life as both a husband and a subject.
Dressed in period costume, Claire is seen asking one advisor, "Don't you get sick of it all, like lonely?"
Her character later admits: "You don't think I would have preferred to grow out of the spotlight? Be away from the scrutiny and the visibility."
A reluctant Prince Philip, meanwhile, is told: "She is the job. Loving her, protecting her. She is the essence of your duty."
Joining Matt and Claire in the period drama is John Lithgow who plays Prime Minister Winston Churchill, and Victoria Hamilton, who takes on the role of the Queen Mother.
According to The Telegraph, a reported $100million has been spent on the period drama, making it the most expensive drama ever made by Netflix.
Written by Peter Morgan and directed by Stephen Daldry, The Crown follows the Queen's life from 1947 to 1956 and is set to air on Netflix this November.