Anime with character that goes to a magic academy

Sounds like: The Misfit of Demon King Academy (Maou Gakuin no Futekigousha)

After 2,000 years of countless wars and strife, the demon king Anos Voldigoad made a deal with the human hero, Kanon, to sacrifice his own life to ensure peace could flourish. Reincarnating 2,000 years later, Anos finds that royal demons now harshly rule over lower class hybrid demons in a society that values Anos's pureblood descendants over the demons who interbred with other species, such as humans and spirits. Finding that magic as a whole has begun to decline and his descendants weaker as a result of the peace he created, Anos, now technically a hybrid himself, decides to reclaim his former title of Demon King, but first, he must graduate from the Demon King Academy where he is labeled a total misfit.

"Mahouka Koukou No Rettousei", English title translation "The Irregular at Magic High School" (weird translation but anyway...). I do not recognize the revival bit, but the rest of it matches. The incident you refer to as "beat the teacher" was a challenge bout with a guy on the student council who thought the protagonist, Tatsuya, was too weak at magic to be worth considering. But Tatsuya is incredibly over-powered in directions that don't show up on their standard tests for magic ability. Considerably later in the series Tatsuya is in an inter-High School mock battle competition where the rules prohibit potentially lethal magic. Tatsuya ampiflies the sound from a finger-snap to deafen and stun his opponent. The bloodline thing is complicated, but what you probably remember is that early on they mention that some families are well known for specific styles of magic and the most [politically] powerful of them adopt family names that contain a number word. Low numbers are generally indicate more political power. Tatsuya's family name is Yotsuba ("Four leaves") but he uses a pseudonym to hide this.

Other recognizable features: Many of the magic users have gun-shaped computers they use to construct their spells before casting. The "gun" is just so that the animation looks dramatic, the shape is not important. There are also sword fights because - hey - anime! Tatsuya's younger sister is actually the more powerful magician but she's mostly in the background.

Like Jake said, it does sound most like "The misfit of Demon King Academy", an anime released a few months ago.

The main protagonist is a demon king reincarnated from 2000 years ago, He is immensely overpowered. In the first episode he uses his heartbeat to injure his opponent and later, clicks his fingers to "kill" his opponent. He then uses unique magic to resurrect and kill his opponent multiple times.

later on in the episode, the character he kills, older brother, who is a Demon Lord (kind of teacher I guess, idk he disappears from the story after that point) wants revenge for the protagonist attacking his sibling and attacks the protagonist, with his brother. The protagonist kills and reincarnates the younger brother as a zombie to taunt/threaten the older brother.

There's also a whole political thing about 'royals' and 'hybrids' where royals are seen as superior and hybrids, worthless/useless.

So yeah, that show pretty much every matches his description.