Eckels is a pompous blowhard. That might be not be technical enough, but it is how I would describe him. Perhaps a better character description of Eckels would be to describe him as an arrogant coward.
Right from the beginning of the story, Bradbury paints Eckels as a guy that the reader takes an aversion to.
The muscles around his mouth formed a smile as he put his hand slowly out upon the air, and in that hand waved a check for ten thousand dollars to the man behind the desk.
He comes walking into Time Safari Inc. like he owns the place. He's waving around his big check like he is something special and better than the "lowly" workers that will be in charge of his life. Eckels is a man who thinks that the world revolves around him. Take this quote for example:
"Shooting my dinosaur," Eckels finished it for him.
Eckels thinks that the entire safari into the past is for him and only him. Keep in mind that there are several other hunters that go back in time with Eckels.
Even once the time machine enters the past, Eckels still has not grasped the danger of the very real situation that he is in. He brags about how many other safaris that he has been on and how many animals that he has killed. He even plays with his gun like it is a mere toy to him.
Eckels, balanced on the narrow Path, aimed his rifle playfully.
When it comes down to it though, Eckels is a coward. Granted, a T. rex would be a scary sight, but Eckels is the only hunter who so completely loses his self control that he is unable to follow a simple path back to the time machine.
"Get me out of here," said Eckels. "It was never like this before. I was always sure I'd come through alive. I had good guides, good safaris, and safety. This time, I figured wrong. I've met my match and admit it. This is too much for me to get hold of."
"Don't run," said Lesperance. "Turn around. Hide in the Machine."
"Yes." Eckels seemed to be numb. He looked at his feet as if trying to make them move. He gave a grunt of helplessness. . .
Travis shook his hand away. "This fool nearly killed us. But it isn't that so much, no. It's his shoes! Look at them! He ran off the Path."