To be at peace with yourself means that you are comfortable with who you are, what you're doing with your life, and how your actions are affecting others. Gene spends much of his time in A Separate Peace struggling to find some semblance of peace with himself.
Gene's relationship with Finny swings from being thankful to have a best friend such as Finny to feeling that Finny is his greatest threat and most significant rival amongst the Devon School classmates. At one point, Gene sees Finny's invention of blitzball and the development of the nightly Super Suicide Society meetings as a plot to prevent Gene from excelling academically.
My brain exploded. He minded, despised the possibility that I might be the head of the school. There was a swift chain of explosions in my brain, one certainty after another blasted-up like a detonation went the idea of any best friend,
Yet, when Finny returns with his leg in a cast and announces that he intends to have Gene participate in the athletics he can't pursue,
I lost part of myself to him then, and a soaring sense of freedom revealed that this must have been my purpose from the first: to become a part of Phineas.
In the end, Gene finds peace with himself in the passage of time and the adjustments that time and maturity bring.
Everything at Devon slowly changed and slowly harmonized with what had gone before. So it was logical to hope that... I could achieve, perhaps unknowingly already had achieved, this growth and harmony myself.