Peter Ivanovich is the kind of best friend you probably don't want to have. The narrator tells us that he's one of Ivan Ilych's two "nearest acquaintances" (together with Fedor Vasilievich [1.19]).
The two men go all the way back to law school and have a long history together. Once Ivan comes to St. Petersburg, he and Peter work together at the Court of Justice. They see a lot of each other socially, and they're both bridge fanatics.
Yet in spite of seemingly being so close to Ivan, Peter Ivanovich doesn't feel all that much when Ivan dies. He just hopes that it might lead to a promotion for his wife's brother. While he does go to Ivan's funeral, it's only because he thinks he's "under obligations" (1.19). Once Peter gets to Ivan's house, all he wants to do is get out as quickly as possible so he can play bridge with Schwartz and his other friends.
Not that the funeral leaves Peter Ivanovich totally unfazed. Seeing Ivan's corpse is an uncomfortable experience, though not because he misses Ivan. Peter Ivanovich just doesn't like thinking about death, disease, or anything unpleasant, because they make it harder for him to be cheerful and enjoy himself. And that's just about all he wants to do.
Normally Peter tries to avoid thinking all of those nasty things like death and dying. When he's reminded of them or when he starts to think that what happened to Ivan might just happen to him, it makes him very nervous and displeased. He positively can't bear to think about his own death. Which is why, during Ivan's service, "[h]e did not look once at the dead man, did not yield to any depressing influence, and was one of the first to leave the room" (1.48).
Although he doesn't show up much after the first chapter, Peter Ivanovich is the central character of that chapter. The first chapter is told from his perspective (though not in the first person sense). It's through his eyes that we first see the false world of Ivan Ilych – his work place, his friends, his family, and home. And by getting to know Peter Ivanovich intimately, we see first-hand what the people in that world are like. We also see what Ivan was like before he was sick. (We are meant to assume that Ivan was similar to Peter Ivanovich in his background, profession, social circle, and character.)
What we're supposed to take away from Peter Ivanovich – and the world he represents – is that he's false. His friendship with Ivan represents one respect in which he's false – the relationship is not deeply felt at all, and evidently devoid of real care. But that's not the only thing.
Almost all of Peter Ivanovich's gestures feel faked; he makes them only because they're the right thing to do in front of everyone else. He crosses himself and bows before Ivan's casket not because of any feeling he has, but because it's what one does . But his fakeness is even more obvious in his expressions of sympathy for Praskovya Fedorovna:
Praskovya Fedorovna recognizing Peter Ivanovich, sighed, went close up to him, took his hand, and said: "I know you were a true friend to Ivan Ilych..." and looked at him awaiting some suitable response. And Peter Ivanovich knew that, just as it had been the right thing to cross himself in that room, so what he had to do here was to press her hand, sigh, and say, "Believe me..." So he did all this and as he did it felt that the desired result had been achieved: that both he and she were touched. (1.29)
The central falseness in Peter's character, however, is his refusal to admit his own mortality; he carefully tries to blind himself to the fact that he's going to die someday. That's tied in with his selfish desire to deal only with what's pleasant to him. Like bridge games. And that particular falseness in Peter sets up the rest of the story. It's that falseness which is at the heart of Peter's (and Ivan Ilych's) whole world, and it's that falseness which will be brutally exposed over the rest of the story by Ivan's own experience of dying.