A moral question -- what is the purpose of life -- asked of the late 19th century Russian bourgeois.
But in general Ivan Ilyich's life went on as he believed life outght to go: easily, pleasantly, decently. 
In them he saw himself, all that he had lived by, and saw clearly that it was all not right, that it was all a terrible, vast deception concealing both life and death. 
The novella is an under-represented form. If you've never read Tolstoy, it's a place to start.
As for the book itself, it's morally serious and artistically well-wrought. I'd take its message to be that to evade thinking about death we waste our lives, but also that it is very hard to understand death as anything but an abstraction until you are facing it. 
Ivan Ilyich had no clear, definite intention of marrying, but when the girl fell in love with him, he put the question to himself. "In fact, why not get married?" he said to himself. 
He and his wife agreed in their aims, and besides that, lived together so little, they became such friends as they had not been since the first years of their married life. 
When there was nothing more to arrange, it became slightly boring and lacking in something, but by then they were making acquaintences, habits and life became rich 
So a month went by, then two. 
The translation team (Richard Pevear, Larissa Volokhonsky) seems prominent in Russian lit. More on history of English translations here:
Lawrence was part of a generation of English writers who were in rapture over “the Russians.” In the early twentieth century, English-speaking readers had been introduced to the Russian literary giants of the nineteenth century, thanks largely to the tireless efforts of Constance Garnett (1861–1946), who nearly went blind translating seventy volumes of Russian literature into English. In an essay titled “The Russian Point of View” (1925), Virginia Woolf tried to account for her generation’s Russomania. Like Lawrence, Woolf describes Russian literature as a kind of primitivist antidote to the strictures of Victorian society. The soul of the Russian, Woolf wrote, was “confused, tumultuous, incapable, it seems, of submitting to the control of logic or the discipline of poetry.” https://www.bookforum.com/print/2801/george-saunders-looks-for-life-lessons-in-russian-literature-24370
Tolstoy wants to depict the interiority of his characters at their most intense moments; the narrative voice switches from direct address to indirect when Ivan Ilyich becomes sick [preface vii]. Cormac McCarthy comes to mind as an opposite pole, committed depiction of extremes but with dispassionate exteriority and a terrible narratorial calm.
brief biography of Tolstoy
In his time considered more a thinker at large and public figure than a writer. That we now think of him as a novelist might be mostly to do with the Russian Revolution and the USSR's cultural policy. In this sense the anti-Flaubert; art as a vector for social and moral change, not readerly pleasure or authorial mastery.
As he set out to improve the world, his ability to perceive it deteriorated. Instead of conforming his ideas to the particularities of existence, he conformed his perception of reality to his vision for the world...Tolstoy devoted himself to activism and spiritual improvement and paid the mental price. After all, most historical leaders write pallid memoirs not because they are hiding the truth but because they’ve been engaged in an activity that makes it impossible for them to see it clearly. Activism is admirable, necessary and self-undermining -- the more passionate, the more self-blinding.