Your first introduction to Hogwarts in Hogwarts Legacy consists of meeting the blunt and off-putting headmaster, Phineas Nigellus Black. He doesn't seem to be the most forgiving when it comes to the Professors at the school, and bans Quidditch at the beginning of the year much to the students' dismay.
Much of his past is initially unknown to us, but as you dig deeper into his life and his career, you are able to unravel a bit of information on his family and history. He is a member of the House of Black, with notable wizards and witches such as Sirius Black, Narcissa Malfoy, Bellatrix Lestrange, and more.
First Meeting Headmaster Black At Hogwarts
As you first arrive at Hogwarts with Professor Fig for your Sorting Ceremony, you meet Headmaster Black for the first time. He asks where Fig has been and why you have arrived so late, since the Sorting Ceremony has already ended. He then takes you into The Great Hall to be sorted despite the late show.
After you are sorted, Headmaster Black announces that the Quidditch season has been canceled. Despite the groans from students, he holds onto this and then tells the students to prepare for classes the next day. This is your first initial and short introduction to Headmaster Black, who you won't see often afterward. Most of the interactions with students and administrative work are put on Professor Weasley instead, as Headmaster Black seems to loathe forming any relationships with his students. He is often unaware of what is going on with his students and coworkers, and isn't much of a support system for anyone.
Headmaster Black's Family History
There's a lot you can learn about Headmaster Black when you visit his office and bedroom. You learn that he was sorted into Slytherin when he was in school, as is most of the Black family members. There's also a letter you can find in here from his wife, Ursula, that discusses his family and children missing him as he is often not around due to his job. They only get to see their father during holidays, showing he has not much of a relationship with his own family as well.
Another note in here hints at the fact that Headmaster Black attempted to change the entrance requirements for Hogwarts, which the Ministry of Magic turned down. It is likely that he wanted to only allow pure-blood wizards and witches into the school, which is a view that the Black family has had for years. This view caused a lot of members to leave the family though, including Sirius Black from the original books and movies. He even eventually disowned his own son, Phineas, due to his differing views on pure-blood families and his acceptance of Muggle-born wizards.
This is most of what we learn about Phineas Nigellus Black in Hogwarts Legacy as Headmaster of Hogwarts. His children eventually attend the school, but it seems that even then, he does not put his familial or work relationships first. He's mostly in it for the prestige that comes with being the Headmaster. Most of the other Professors show distaste for him, including Deputy Headmistress Professor Weasley, with many choosing to avoid him when possible.