So, Snape killed Dumbledore but he didn't cause his death. The curse and the potion did it. Nobody forced Dumbledore to put on the Ring neither to drink the potion. Dumbledore caused his own death and Snape just finished his life.

Did Snape really betray Dumbledore?

In return, Snape secretly allied himself with Dumbledore and the Order of the Phoenix as a double agent against Voldemort, using his powers of Occlumency to hide his betrayal from Voldemort. ... Even with his efforts to protect her, Snape felt responsible for Lily's death at Voldemort's hands.

Did Dumbledore really die?

As he was about to die by a cursed ring, Dumbledore planned his own death with Severus Snape. Per the plan, Dumbledore was killed by Snape during the Battle of the Astronomy Tower. ... Dumbledore is the only Headmaster who was laid to rest at Hogwarts. A portrait of him still remains at Hogwarts.