There are sadly so many ways to answer this question. Perhaps, one way in which Eliezer died was with the death of his faith, something he was relegated to questioning and denouncing with the atrocities he endured and viewed. Another form of death was the severance to his community, as the Jewish people of Sighet were forced into camps and never were able to live together as a community. Yet one other form of death could be the separation of his family, with Eliezer never seeing his mother and sister again. The more painful experience on this end would be with his father, a situation where Eliezer had no choice but to watch his father die and not be able to do anything about it. The last words on his father's lips was the son's name, who did not respond to it. Another level of death could be the idea of a child who loved life and everything in it maturing into a young man who could only think of surviving. In this light, Eliezer's death is the death of his soul so that he is almost animalistic in his pursuit of survival.
Imagine what people had to endure and survive during the Holocaust - the things they had witnessed, the losses they had suffered and often before their own eyes, the long term physical and emotional abuse, physical starvation, hard labor.
Any psychologist will tell you that it is virtually impossible to come through such events unscarred, and that was true of Elie as well. He shut off a part of his brain, his emotion and his being in order to survive, and having survived it, there was some of that humanity he could never regain. A part of his life was permanently destroyed, as it was in everyone who survived the Holocaust.
Elie Wiesel, like many of the other survivors of the concentration camps, were physically starved. Their figures were nothing more than a skeletal frame with skin on it when the camps were liberated. In this sense Eli physically resembled a corpse.
Mentally more things than just Elie's body died. His faith in God died away as he watched over and over the horrors around him, experienced the inhumane treatment at the hands of the Nazi's, and found no comfort from God. He had been very spiritual before and longed to study more and more about God, but after watching the young innocent boy hang, he stopped believing in God because he could find no answers.
Elie's role as a son also died away. He had two parents and relatives when he went into the camps. When he came out his family had all been murdered. In addition, he felt he had lost himself as a son to his father because he had not stood up and helped his father and at times wished he were not there to burden him.
Elie also died because he was no longer the person he had been before he went into the camp. His experiences had changed him so dramatically and the changes of no longer having his family or a place in his homeland left him with the need to rebuild his life and self as a concentration camp survivor.