Why didn't Harry die when he killed Voldemort?

No. Voldemort's soul went inside Harry when he killed Lily. When Voldemort made the new body with Harry's blood, he was connecting himself to Harry but not necessarily the other way around. The connection already existed through harry being the horcrux. And that connection was destroyed before Voldemort's final curse. Whether the connection was destroyed in the forest or when Voldemort cast the curse with Harry's wand is not made clear, but I suspect the former. He certainly had no reason to speak serpent after the Killing Curse in the forest.