I think, in answering this question, it's useful to consider how this story is structured, because The Death of Ivan Ilyich begins at the end, after its protagonist has died, with Ivan Ilyich's funeral. In a way, then, it might be useful to consider the funeral scene itself and how it relates to the larger story that follows.
In the funeral scene, what we see is a focus on appearances. Ilyich's friend Ivanovich (whose viewpoint we initially follow) is largely focused on giving an appropriate impression. You see this in his continued observance of the signing of the cross (which is joined with his internal preoccupation concerning the appropriateness of the gesture) or in his greeting the widow, Praskovya, who is herself primarily interested in maximizing the amount of money she can receive from the State, now that her husband is deceased. In reading this scene, you might get a sense of artificiality on the part of its participants and, ultimately. dishonesty.
From here, we follow Ivan Ilyich, starting in his youth and carrying on until his death, mirroring the funeral scene itself. We find that for much of his life, Ilyich had possessed a single minded focus on appearances. However, as his illness takes root, he comes to realize just how hollow (and ultimately meaningless) his entire life had been, and as his illness worsens, his despair only grows. What we see in Ivan Ilyich is a condemnation of the materialistic lifestyle, which provides no comfort in the presence of death.