I would not describe Eckels in very positive terms. I have never liked the character of Eckels. The fact that he messes everything up in the present doesn't exactly endear him to me.
I would mainly describe Eckels as arrogant and selfish. My feelings about arrogance start right from the beginning of the story.
"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."
Apparently time travel isn't cheap in the year 2055. No problem there. But most people would approach the desk and present their check. Not Eckels. He thinks he has to wave his money around for all to see. Like it's a banner of privilege and status or something.
I get feelings of selfishness from three specific lines. First, this one:
"Does this safari guarantee I come back alive?"
Notice that Eckels wants a guarantee on his life. I understand that, but I get the feeling that Eckels only cares about his life and not the lives of the other hunters and guides. He could have asked "Does this safari guarantee we come back alive?" The sentiment would have been the same, but Eckels wouldn't appear like a selfish rich kid.
The second line of selfishness is this line:
All you got to worry about is-"
"Shooting my dinosaur," Eckels finished it for him.
Eckels is so focused on what's his that he even has to interrupt the Safari Inc. employee.
The last support for Eckels being selfish (and a bit ignorant) is when the rules are being explained to him about the possible repercussions of time travel. Eckels just doesn't get it and/or doesn't seem to care.
"And all the families of the families of the families of that one mouse! With a stamp of your foot, you annihilate first one, then a dozen, then a thousand, a million, a billion possible mice!"
"So they're dead," said Eckels. "So what?"
Every time that I read Eckel's response, I hear him speaking the line in a confrontational tone. I don't hear him asking an actual question of concern. Eckels genuinely doesn't care if those mice are dead, because he is not a mouse. It doesn't concern him, so he doesn't care about those lives. If he doesn't care about those lives, why would he care about the lives of the other members of the time travelling group? Eckels is an arrogant and selfish rich brat.