Kurt Vonnegut, Jr. writes this dystopian short story with a very wry, dry sense of humor in a matter-of-fact and straightforward way. For example, take a look at the first line: "The year was 2081 and everyone was finally equal." That is very factual and dry, but the addition of the word "finally" is very sarcastic; it implies it was what everyone's goal was all along, and that it was indeed possible. He writes with that same sardonic tone throughout, and it adds a feeling of humor and sad derision. Take for example, a line soon after the one listed above:
"Some things about living still weren't quite right, though. April for instance, still drove people crazy by not being springtime."
He comments on that darn April, that just won't stay in line like everyone else has. He is trying to be funny, but to express the seriousness of the actual society, because they really think that way. It is using satire to point out the absurdity of having a society that is truly equal through artifical and enforced means. Even though we know that April can't be tamed, he indicates that the people in the story feel it should be. We feel a gap in perspective there, which lends itself well to the bemused tone of the piece.
So, through the use of sarcasm and black humor, and a dry, straight-forward tone, Vonnegut creates a tone of serious mocking and storytelling wit. I hope that helps a bit; good luck!