The Kovarian Chapter was a rogue faction of the Church of Silence led by Madame Kovarian, which included several Silents.
History
Circa the 52nd century, (TV: A Good Man Goes to War) the Time Lords sought to return to the universe. They enacted a plan to safely stage their return, sending a signal through a crack in time on the planet Trenzalore, hoping to contact the Doctor.
Tasha Lem, Mother Superius of the Church, granted the Eleventh Doctor exclusive access to Trenzalore. Upon discovering the crack in time on the planet's surface, the Eleventh Doctor decoded the signal and discovered it was actually a question: "Doctor who?". By speaking his name through the crack in time, the Doctor could give the Time Lords a 'green light' to return. Much of the universe was soon thrown into panic, fearing the Time Lords' return would reignite the Time War. Despite the Doctor vowing to never speak his name, various hostile forces commenced a siege on the planet, hoping to stop the Time Lords' return.
The Church assisted the Eleventh Doctor with protecting the planet and its inhabitants for over one thousand years, dedicating themselves to ensuring 'silence will fall'. However, the Church's Kovarian Chapter broke away and became a rogue faction. They dedicated themselves to killing the Doctor, which would conclusively ensure 'silence will fall', and prevent the Time Lords' return. The Chapter travelled back in time, resolute on killing the Doctor before he ever arrived at Trenzalore. (TV: The Time of the Doctor)
The faction made two (seemingly separate) attempts to kill the Doctor, the first by taking control of his TARDIS and causing it to explode on 26 June 2010. This, however, resulted in a total event collapse that, although corrected, left the cracks in the skin of the universe in the first place. This attempt was a complete failure, causing the very problem they were attempting to avert which created a temporal paradox. (TV: The Pandorica Opens)
The second attempt involved the Silents influencing human history for thousands of years, culminating in the creation of an astronaut suit that would be operated by River Song, whom they kidnapped and raised to kill the Doctor (TV: A Good Man Goes to War) at Lake Silencio, Utah. Because it was a still point in time, they hoped to create a fixed point for the Doctor's death. (TV: The Impossible Astronaut/Day of the Moon) However, River refused to kill the Doctor, resulting in all of time happening at once, before this was corrected by the Doctor faking his death with the Teselecta, making this attempt another massive failure, as the Doctor learned of Trenzalore from Dorium Maldovar. (TV: The Wedding of River Song)
The Doctor later commented that the Kovarian Chapter was caught in a "destiny trap," saying that you couldn't change history if you were part of it, since destroying the TARDIS created the crack through which the Time Lords could return, which they wanted to prevent. Furthermore, the Doctor may have never reached Trenzalore, if the Kovarian-influenced River Song wasn't there to help him in his adventures. (TV: The Time of the Doctor)
Ultimately, only Kovarian herself remained after all of their failures; unable to see that the Doctor would triumph no matter what she schemed.
Not knowing the damage to the universe she would wrought by killing the Doctor too early, (TV: The Name of the Doctor) Kovarian set out to kill the Fifth Doctor with a clone of River acting as a sleeper agent while posing as a companion. While she initially did succeed, the clones turned against Kovarian and saved the Doctor, leaving her trapped in an electric forcefield as punishment for ruining River's childhood and nearly destroying the universe herself. (AUDIO: Furies)