I made up my mind to take the life of the old man, and thus rid myself of the eye forever. (2)
The fact that the man is old, and probably near death anyway, makes this bitterly ironic. To die is one thing, but to grow old, to spend the final years hated and despised is another.
All in vain; because Death, in approaching him had stalked with his black shadow before him, and enveloped the victim. (5)
This is a curiously structured sentence. At first it seems the narrator is simply personifying death. He's also referring to himself as Death, with a capital D – the nemesis of the "Evil Eye." The narrator is the stalker, and his shadow is black (because there is no light; this is before the opening of the lamp).
Yes, he was stone, stone dead. (7)
The word "stone" is repeated a third time, a few sentences later, which suggests that the word is important. This makes us think of the Medusa myth. When a person looks at Medusa and she looks back, or vice versa, the person turns to stone. This is just the kind of thing that would appeal to our narrator.