There are many quotes that illustrate Gene's characteristics of competitiveness, insecurity, or jealousy. Indeed, a major aspect of the story is the war Gene sporadically conducts against himself, Finny, or others at Devon School.
Consider the Upper Middle Class term tea. As Finny, Gene, Mr. Patch-Withers, and Mrs. Patch-Withers are discussing the actions of the Germans, Finny unbuttons his sportcoat so as to have freedom of movement for larger gestures. The coat falls open and exposes Finny's belt, his school tie with Devon's crest and colors. Gene finds himself looking forward to observing Finny's punishment for this misstep, then being jealous when no punishment is given.
This time he wasn't going to get away with it. I could feel myself becoming unexpectedly excited at that...Phineas was going to get away with even this...He had gotten away with everything. I felt a sudden stab of disappointment. That was because I just wanted to see some more excitement; that must have been it.
The study scene in their dorm room displayed all the traits you ask about. Gene realizes he wants to achieve scholastic recognition to compare with Finny's athletic honors; Gene is too insecure to conceive of a friendship with someone by whom he feels (at that moment) betrayed; and Gene is jealous that Finny's sports achievements come so easily, while Gene has to study and work for his good grades.
"Somebody's got to be the head of the class."...He had won and been proud to win the Galbraith Football Trophy and the Contact Sport Award...If I was head of the class on Graduation Day and made a speech and won the Ne Plus Ultra Scholastic Achievement Citation, then we would both have come out on top we would be even, that was all..."You wouldn't...mind if I wound up head of the class, would you?"..."I'd kill myself out of jealous envy." I believed him...My brain exploded. He minded, despised the possibility that I might be head of the school.