Why Coriolanus is a tragic character?

Coriolanus is a tragic hero because he meets all the criteria.

First, he is a man of elevated birth.  He comes from one of the finest families in Rome.  He proves himself as a hero and savior of Rome in his battles against the Volscian.

Like many tragic heroes before him, Coriolanus is guilty of hubris.  His pride will not let him "lower" himself.  He considers himself better than the plebeians who question him.

His excessive pride leads him to abandon Rome and his family and join the Volsci.

Once he turns his back on Rome, he is doomed.  That he must die in the end is a foregone conclusion.

In the end, it is his pride that causes his downfall and that makes him a tragic hero.