The most prominent reason that Tolstoy's "The Death of Ivan Ilyich" speaks to his life more than his death is that in death, people tend to reflect on the lives of the person who has passed away.
In the opening of the text, Ilyich's colleagues are shocked to find out that Ivan has passed away. After their initial surprise subsides, they begin to discuss the advantages of Ivan's death (in relation to his newly opened position).
The remaining chapters are spent recollecting Ivan's life (his youth, rocky marriage, and turn in health). These chapters offer the reader insight into Ivan's life and his death.
The story focuses upon Ivan's life, more than his death, because death is considered an end. If Tolstoy were only to focus upon Ivan's death, the story would be a short one and fail to defined why the death of Ivan is important enough to tell (through the telling of his life).