I'm guessing this might be Assassin's Pride.
In a world where only nobles wield the sacred power of mana to wage war against the creatures of darkness, a curious absence of ability places young Melida in a precarious position. Under the guise of tutelage, Kufa, a ruthless assassin, is sent to assess Melida’s abilities and determine if she is the true-born heir of the prestigious Angel family. If she does not measure up to her family’s expectations, another, more ruthless plan will be set in motion…
Kufa is half-vampire.
Kufa Vampir is the main protagonist of the series Assassin's Pride. He is half human and half vampire, the strongest of the Lancanthrope class. He also belongs to the Guild Jack Raven.
A search for
anime magic tutor vampire brought me to this page, and Assassin's Pride looked the most likely, so I searched for more details, which brought me to the wiki.