In 1995,  Dolores Umbridge, now in Azkaban for crimes against humanity, had joined Cornelius Fudge's team. Umbridge  was an anti-Muggle-born and anti-part-human, corrupted Ministry officer who weaseled her way into a position of power at the Ministry.

Fudge had turned against Harry Potter after the events of the Triwizard Tournament, and was only happy to press bogus charges against Potter when he summoned a Patronus to save his own and a Muggle relative's lives. What Fudge didn't know-Umbridge had sent dementors after Potter to create this opening. This was the beginning of Potter and Umbridge's long history of enmity.

At the time, Fudge had an irrational fear of Dumbledore's creating a secret army. Emboldened by Umbridge, he sent her to complete tasks for him at Hogwarts. She was to assume the Defense Against The Dark Arts teaching position, keep an eye on Potter, and prevent the students of Hogwarts school from learning any defensive magic so Dumbledore couldn't 'gather them for his army.'

Once at Hogwarts, Umbridge did exactly that. This infuriated the students at Hogwarts, particularly Potter.
Inspired by the need to learn Defense Against the Dark Arts during a dark time for Hogwarts, Hermione Granger proposed to undermine Dolores Umbridge and create a secret study group of sorts. This was the beginning.