Hermione Granger has been alive throughout the entire Harry Potter series, including Harry Potter and the Cursed Child. Therefore, no one killed Hermione. However, in Harry Potter and the Cursed Child, there exists an alternate reality in which the Augurey rule the world after Harry Potter's death and Hermione is a rebel fighting against them. In this reality, Hermione is killed by the dementor's kiss along with Ron. Technically, even this is not death as the dementor's kiss separates the soul from one's body, but this is the closest Hermione comes to getting killed in the Harry Potter series.