This very popular myth projects Shakuni as some kind of mastermind who had taken a vow to destroy the Kuru dynasty and it was his machinations that led Duryodhan to the war, which ended up destroying the entire Kuru dynasty. 

There are various versions of this myth but one of the most popular ones is that when Bhishma wanted to marry Dhritarashtra to Gandhari, nobody in the Gandhar royal family was happy about it because Dhritarashtra was blind and  not the King. When they opposed, Bhishma captured all of them and kept them as prisoners. They were supplied food for only one person everyday. All the members of the family decided to offer that food to Shakuni and starved to death. Shakuni in turn took the vow to destroy the Kuru dynasty to avenge the destruction of his family at the hands of Bhishma. 

I am not sure about the origin of this myth but Ved Vyas’s Mahabharata makes no mention of this. While Shakuni always took part in Duryodhana’s evil plans along with Karna and Dushasana, he did not come across as any kind of mastermind. In fact Shakuni comes across as someone who loved his nephew and wanted his welfare. 

In Mahabharata after Ghoshyarta where Pandava saved Duryodhan from Gandhrava’s Shakuni advised his nephew to forget the animosity against his cousins and told him to look for peaceful ways of coexistence. 

Regarding Shakuni’s family being killed by Bhishma, there are mentions of Shakuni’s brothers fighting alongside him in the Kurukshetra war, hence we can safely conclude that this story is a myth and is not supported by the original Mahabharata. 

One example from Vyasdev’s Mahabharata below establishes this without doubt that Shakuni’s brothers were part of Kurukshetra war and were not killed by Bhishma in some slasher movie inspired scheme.