Gene does demonstrate some rather unpleasant traits throughout the course of the novel; he shows envy, pettiness, hatred, dishonesty, cowardice and betrayal. And throughout it all, he shows a definite awareness of his poor qualities, and struggles with his own distaste for himself.
In the beginning of the novel, he feels bitterness and sheer envy for Finny. After Finny, on the beach, reveals that Gene is his best friend, Gene doesn't return the compliment. He keeps his mouth shut, and later pettily concludes that Finny is trying to undermine his studies. This episode reveals how insecure and self-centered Gene is. When Gene goes to Lepers house, and is confronted by Leper's startling change in personality, Gene ends up yelling at him, calling him names, and fleeing the scene instead of being the kind, supportive friend that Leper needed in his time of struggle and confusion. In chapter seven, when Brinker half-jokingly confronts Gene about what happened at the tree, Gene outright lies, and to divert the attention away from himself, picks on another, lesser-liked kid in the room. He belittles and makes fun of the other kid, all to save himself from the discomfort of being confronted about Finny.
Finny does not have quite as impressive resume of negative traits as Gene, but he does have a couple moments. He lies about enlisting in the war and wanting to be involved, and he is a bit self-centered, designing activities and games that give him the advantage in all things. His own denial of the accident contributes to awkwardness in his friendship with Gene, and inadvertantly, to his anger at the end that led to his fall. He didn't want to face the facts, and accept any form of betrayal into his life.
All of these instances, and more, show Gene in a not-too flattering light, and give him depth and make him seem like a real person. Finny himself has his moments. We watch throughout the entire novel as Gene and Finny struggles against their own vices and insecurities. I hope that those thoughts helped; good luck!