He figured if he worked at it – and Lord, had he worked – he could be the fastest runner in the fifth grade when school opened up. He had to be the fastest--not one of the fastest or next to the fastest, but the fastest. The very best. (1.7)
Jess's desire to be "the fastest" runner is practically the first thing we learn about him. We know right away that he has talent and determination. We also see him trying to carve out a place for himself and achieve something at school, and really become known for something: for being "the very best." That desire to be the best says good things about his character and his moral state.
One time last year Jesse had won. Not just the first heat but the whole shebang. Only once. But it had put into his mouth a taste for winning. (1.18)
This is the root of Jess' ambition: winning. The "taste of winning" – of that short-lived glory – inspired him to train all summer and find a place for himself in the school hierarchy the following year by being the fastest runner. It gave him purpose and direction. Even though it happened "only once," it was enough to make him completely motivated to win the following year.
"All right, Jesse. Get your lazy self off that bench. Miss Bessie's bag is probably dragging ground by now. And you still got beans to pick."
Lazy. He was the lazy one. He gave his poor deadweight of a head one minute more on the tabletop. (1.63-64)
This is all about context. We readers know Jess isn't lazy and so we side with him. We saw him get up and train for his race while the whole house was still asleep, and we just witnessed his older sisters bamboozle their mother into letting them get out of chore duty. Despite that, Jess is the one who gets called lazy and who has to do the most around the house. It just doesn't seem fair.
This was the day he was going to be champion – the best runner of the fourth and fifth grades, and he hadn't even won his heat. (3.73)
Humiliation ensues. Jess worked all summer for this – we saw in the first chapter how hard he trained, and how tired it made him – and now he doesn't even make it to the final round. The story he had written for himself about how this "day" and this year would go is instantly unwound – instead of being fastest, he's not even in the top four.
Jess knew now that he would never be the best runner of the fourth and fifth grades, and his only consolation was that neither would Gary Fulcher. They went through the motions of the contest on Friday, but when it was over and Leslie had won again, everyone sort of knew without saying so that it was the end of the races. (4.2)
It's all over for Jess's dream of being "the best runner" by the end of fifth grade, and it's all because he stood up for a girl and for equality. He was the one who fought for Leslie's ambition to run in the race and, because of that, wasn't able to fulfill his own ambition of winning himself. What's sad about this is that Leslie wins so definitively that no one else takes pleasure in racing or competing anymore.
How could he explain it in a way Leslie would understand, how he yearned to reach out and capture the quivering life about him and how when he tried, it slipped past his fingertips, leaving a dry fossil upon the page? "I just can't get the poetry of the trees," he said. (4.111)
With the excitement of the races dissipating, and with his new friendship with Leslie strengthening him, Jess returns his attention to drawing. His ambition to draw well is as strong as his desire to run fast. Here, though, he seems to get results less quickly. He knows what he wants to do but can't seem to achieve it.
The stream was a little lower than it had been when he had seen it last. Above from the crab apple tree the frayed end of the rope swung gently. I am now the fastest runner in the fifth grade. (12.21)
Numbly, Jess reflects on his achieved and once-longed-for ambition. It's morbid, but he's at the very place where Leslie died, looking at the rope that failed her when she was crossing the creek. Implicitly, we understand that Jess feels like he might have failed her too, by not being there to convince her to wait and go to Terabithia only when the creek calmed down. Instead of thinking about those things, he turns his attention to the fact that he's become the "fastest runner."
He was suddenly ashamed that he'd thought he might be regarded with respect by the other kids. Trying to profit for himself from Leslie's death. I wanted to be the best – the fastest runner in the school – and now I am. (13.45)
This isn't how Jess wanted to win that race or become "the fastest runner" at all. Now he has what he wanted at the very beginning of the book, but it's meaningless. He's the fastest runner now, but it's not because he improved, or trained harder, or got fancy new running shoes. It's because his competition can't race against him anymore.
Now it was time for him to move out. She wasn't there, so he must go for both of them. It was up to him to pay back to the world in beauty and caring what Leslie had loaned him in vision and strength. (13.60)
By absorbing Leslie's ambition into his own, Jess's desire for shaping his future life becomes deeper and more profound. It's no longer about being just the fastest runner in fifth grade, or creating awesome drawings. It's about making a lasting contribution to the world that reflects what Leslie taught him and helps memorialize her in another way. Instead of striving for records or material things, he will "pay back to the world in beauty and caring what Leslie had loaned him in vision and strength." Because she can't keep on giving that "vision and strength," he'll take on the responsibility of turning those qualities into "beauty and caring," and send them back out into the world.
They gave Jesse all of Leslie's books and her paint set with three pads of real watercolor paper. "She would want you to have them," Bill said. (13.63)
Even though Leslie's no longer there, she's still able to give Jess one last push of encouragement. Through her parents' gift of resources and supplies, she encourages Jess to keep learning, thinking, and drawing. While Jess doesn't have Leslie to tell him stories, he'll have her books, where she found those stories, to inspire him.