Symbolism, Imagery, Allegory
James uses his eyeballs as a barometer for his recovery. His eyes signify how his health is coming along, and they also symbolize how he's able to accept that fact that he is a recovering addict.
The first time James looks into the mirror, he can barely bring himself to really look at his own reflection: "I want to look beneath the surface of the pale green and see what's inside of me, what's within me, what I'm hiding […] I try to force myself but I can't" (1.5.7). Later, he says: "I want to look into the self that lives beneath. Closer, closer. I can't do it. No f***ing way" (1.8.18).
James is terrified by the prospect of really looking at who he is on the inside. Can you blame him? Most other people don't want to have to look at him either: "I might never look into the pale green of my eyes again. There are places from which you cannot return. There is damage that can be irreparable" (2.1.48).
As James nears the end of his stay at the clinic, his eyes have finally started to look healthy, and he's finally able to look deep into them without blinking: "The whites are white and lined with the pink run of thin veins. I follow the veins to the edge of the green. It is a faded olive with small dots of brown…" (3.4.226). He keeps describing his eyes until he gets to "the black where all is revealed" (3.4.226).
The idea is that James is finally able to look at life with clear sight, now that he's gotten himself out of addiction. He's also finally able to see himself clearly and to understand his weaknesses and strengths. These two things are related—James isn't able to move on with his life until he accepts both life for what it is and himself for who he is.