When Rainsford first meets Zaroff, Zaroff knows exactly who Rainsford is. Although that's just a little creepy, it does indeed reveal his character. This shows he is well-read within the field of hunting.
Furthermore, Zaroff is wealthy. This island seems to be his very own. Whether he built the chateau on his own, or it came with the island, his imported supplies reveal that he can afford to keep living this way or he does it quite unethically. In the text, look for the description of the scene when they eat together. This demonstrates the core of his wealth.
Finally, Zaroff needs challenge. He expresses to Rainsford why he hunts man without any reservation, he is bored. When Rainsford climbs the tree and Zaroff knows that he is up there, he blows the rings of smoke to reveal that boredom once again. He did not want the game to be over at that point.