What are the character's situations in the exposition "A Sound of Thunder" by Ray Bradbury?

The word exposition comes from Latin and means "a showing forth." It is the part of the story, usually at the beginning, that introduces background information on the setting, characters, and events in a story. 
In Ray Bradbury's 1952 short story, "A Sound of Thunder," events center around the main character, known only as Eckels. In the exposition of this story, Eckels has come to a place known as "Time Safari, INC." Eckels pays ten thousand dollars to go on a time-traveling safari to shoot a Tyrannosaurus Rex. The company, Time Safari, INC, is described as a vast office with a time machine that glitters and glows like an aurora. His safari guide is Mr. Travis, and he will be traveling back to the late Cretaceous period, about sixty-five million years ago. The story is set in the year 2065, which is evidenced in this quote: 

"First a day and then a night and then a day and then a night, then it was day-night-day-night-day. A week, a month, a year, a decade! A.D. 2055, 2019, 1999, 1957, Gone!"

Lesperance is the man in charge of setting up the safaris in a way that will not disturb the past. His job is to travel back in time, find animals that are about to die, mark them with a red paint bomb, and schedule the time machine to arrive within a few minutes of their natural deaths, so that the chain reaction of the hunt will not disturb the future events. 
The company Time Safari, INC, has created metal paths that are anti-gravity and float above the earth so that hunters will not disturb even the grass. 
The discussion in the beginning of the story mentions the recent election, and how fortunate they all were that President Keith was elected. His opponent, Deutscher, was a dictatorial type who was anti-everything and surely would have ended time travel. 
Eckels seems motivated to hunt by the bragging rights of saying he's taken down a dinosaur. He has shot ever other kind of modern big game and is looking for the next thrill.