Your question had me stumped for a bit, because I didn't remember Eckels being playful. I did remember him being scarred out of his mind when he finally saw the Tyrannosaurus rex. From that point forward, Eckels was so scarred that he could barely follow simple directions from Travis.
But after going back through the text, I noticed a different personality in Eckels during the first half of the story. He is full of bluff and bluster. He is supremely overconfident in his skills and abilities.
“I’ve hunted tiger, wild boar, buffalo, elephant, but now, this is it,”
Once arriving in the past, Eckels is still completely of the mindset that shooting a dinosaur is going to be child's play.
Eckels, balanced on the narrow Path, aimed his rifle playfully.
He is holding his rifle up to his shoulder and playfully taking aim at various imaginary targets. Travis gets mad at Eckels right away, because he doesn't want the gun going off accidentally. That might kill something in the past that shouldn't be killed and wreck the entire future.