This is a tough question because Gene sees himself differently at different places in the novel. That fits the novel's theme of identity as well. Gene struggles to figure out exactly who he is, and his relationship with Finny really messes with Gene's ability to find his own identity apart from Finny. If I had to pick a single way in which Gene sees himself, I would say that he sees himself as a "deceiver." That's not exactly high praise, but I think it fits. He deceives Finny about their friendship throughout the novel, and he deceives the others about what happened on the tree branch. In chapter 12, readers can find a good quote that highlights this trait being present in Gene.
None of them ever accused me of being responsible for what had happened to Phineas, either because they could not believe it or else because they could not understand it. I would have talked about that, but they would not, and I would not talk about Phineas in any other way.
Gene isn't quite sure the exact reason why the other boys believe his tree branch innocence, but Gene is aware that he has sold them the idea that it was an accident. I also like the following quote for Gene being a deceiver:
Phineas was a poor deceiver, having had no practice.
The quote is intended to describe an aspect of Phineas; however, it also tells readers an important detail about Gene. Finny is a poor liar because he doesn't get practice, but this quote seems to suggest that Gene is a good liar because he does get practice. It's for this reason that Gene is able to easily identify Finny's lack of talent in this particular area.