Clementine changes hair color because she's crazy and fun and spontaneous and, like, sooo random.
Well, okay—we're kind of kidding. But not really. Clementine wouldn't have colored hair if it wasn't at least a little bit symbolic, and that's what we get here. We're supposed to see her the way Joel sees her, which is as one crazy cat who has come to make his life better.
But Clementine hair is actually more than that: it's also way for us to keep track of her chronologically. Clem's hair color becomes a big clue as to where in time we are at any given scene. We know when her hair is blue that we are in the present; when her hair is orange, or a deep reddish auburn, we know that we are in Joel's mind; and when her hair is green, we know we're witnessing Joel's memory of the first time he met her.
The color of Clementine's hair also represents the status of her relationship with Joel. The deep reddish auburn color represents the happy days of the relationship. This is her hair color when they delve into Joel's past, when she becomes Mrs. Hamlyn, and when they're hiking in the woods or lying on the ice.
When her hair turns lighter to match the color of her sweatshirt (the color she would probably call Agent Orange) we understand that the relationship has started to go south. Her hair is this color, for example, when she and Joel argue at the flea market, at home during Halloween, and when she comes in late.
When Clementine's hair turns blue in the present, it's a fresh beginning of sorts, a blank slate. At least that's the happy interpretation of it. The color is called Blue Ruin, which could be a reference to cheap gin or just hint at ruin in general. (Blue ruin was a cheap kind of gin that was tinted blue and would, presumably, ruin you. It was often homemade so you didn't know exactly what you were gonna get.) Clementine's hair is blue when she experiences her most intense moment of identity loss, when her mental state is seriously in ruins.
Finally, the green hair at the end (or the beginning), when Clementine and Joel first meet, seems to represent the beginning or birth of the relationship: it's a springtime color. We also know that its name is Green Revolution, so it could represent a revolution, a great change in Joel's life. It could also refer to a revolution as a cycle, hinting at the repetitive nature of Joel and Clem's relationship.