He is a member of Salem's group.
Hazel is Gretchen's older brother. Hazel cares deeply for his deceased sister and her death at Beacon Academy is his reason for wanting Ozpin dead. When she was alive, Hazel was against her joining Beacon to become a Huntress. Gretchen's death had a profound impact on her brother, leaving him a broken and vengeful man who is clearly still grieving despite the untold number of years since her passing.
He is loyal toward Salem and dutifully follows her orders.
His only acknowledgment of Cinder is to speak ill of Cinder being thwarted by Ruby Rose, though in doing so he also acknowledges her as a peer.
Ozpin informs Oscar in "Kuroyuri" that Hazel is someone from his past and advises Oscar to be cautious of him as a result. In "Vault of the Spring Maiden", the reason why he wants Ozpin dead is because he blames him for his younger sister's death. His vendetta against Ozpin being of such magnitude that he feels the need to kill Oz in every incarnation that he can possibly get his hands on, determined to make him suffer over and over again.
When Oscar has trouble getting a train ticket out of a machine at a train depot due to lack of funds, Hazel shows up and bangs on the machine. This causes a train ticket to fall out of the slot. Before moving on, Hazel advises Oscar not to let a small obstacle obstruct his path.
When they meet again in Haven, Hazel is shocked to learn that Oscar is the new incarnation of Ozpin. Though he attacks him, he says his blood will be on Ozpin's hands, and hopes Oscar can forgive him for seeking to kill him.
The pair are seen meeting with Sienna Khan in "Dread in the Air", to gain her loyalty to Salem's Faction. However, Adam enacts a coup and overthrows Sienna as High Leader, a plan Hazel knew nothing about. Hazel is upset with Adam keeping secrets from him, along with killing Sienna when he deemed it unnecessary.
The lack of cooperation between them is shown again in "Downfall", when Hazel refuses to help Adam after he and his forces are confronted by the Belladonnas, armed citizens of Menagerie, along with the Mistral Police.