One point that relates to the idea that the only enemy is really found in the self comes when Gene finds that he has no enemy. This happens when Finny begins training Gene for the Olympics. At this point, Gene is at his happiest. This seems to be true because Gene is no longer in opposition or competition with Finny but is instead joined with Finny in a single purpose.
In addition to these other comments, consider how Gene develops in his self-awareness, and in particular how his relationship with Finny in some ways actually improves after he knocks him off the tree and he is crippled. It seems as if Gene has become aware of his own greed, envy and desire for success and how he hated Finny to some extent for his natural athletic ability, confidence and poise - traits that Gene never had. This to me is a key turning point in the novel of self-awareness - Gene learns an important lesson about his character and his humanity, which compares him starkly with the innocence that Finny displays.
In addition, you may also wish to consider these words of Gene,
...and all the other buildings and all the people ther [at Devon] were intensely real, wildly alive, and totally meaningful, and I alone was a dream, a figment which had never really touched anything. I felt that I was not, never had been, and never would be a living part of this overpoweringly solid and deeply meaningful world around me. (Chapter 12 after Gene visits Phineas)
'No, I don't know how to show you, how can I show you. It was just some ignorance inside me, some crazy thing inside me, something blind, that's all it was. (Ch. 12)
Because it seemed clear that wars were not made by generations and their special stupidities, but that wars were made instead by something ignorant in the human heart. (Ch. 13)
Gene changes when he reaches his moment of truth, his epiphany, in which he realizes the reason for his actions, the reasons for wars, the reasons for people's fears, the reasons why people build their "Maginot Lines.": the "something ignorant in the human heart." They are afraid. But Phineas has never been afraid. Because he has never been afraid, Finny has not felt the need to boast, or to compete, or to achieve some notoriety, or to defeat his friends.
In connection to Gene's inner struggle and his ultimate realization that he was battling himself all along, discuss Gene's quote on the last page of the book in which he states that he was on "active duty all [his] time at school" and that he knows that he "killed [his] enemy there." Right after this, he discusses how Finny is the only character who didn't seem to be at war or afraid and makes it clear that he (Gene) once saw himself as his own enemy.
While Gene was at Devon, he lived in great fear--mostly the fear that he was somehow inferior to others, not good enough. This made him very insecure and competitive. He always had to prove himself, mostly to himself. (Being an excellent student was one way he bolstered his own self-esteem.) He became jealous of Finny because he felt so inferior to him. By the end of the novel, Gene has realized this truth about himself and understands why he made Finny fall from the tree--an act of jealousy and blind destruction.
Gene also realizes by the end of the story that all the boys at Devon lived in fear and insecurity, not just himself. Fear of the war haunted them all, except in Finny's case. His worst fear was being left behind because of his injury. Gene finally realizes that the part of him that struck out to destroy Finny is the same human impulse that causes war in general. He has learned something very important about man's basic nature.