The boys in The Outsiders wore long, greasy hair as a reflection of their rebellious natures and to brand themselves as "greasers"; this is the "trademark" and the "one thing [they] were proud of." Although the boys may not be able to afford the attire or lifestyle of the rich, their hair acts as an assertion of their own brand of social status. By imitating the "tuff" hair of Elvis Presley, the greasers solidify their sense of style and identify themselves as part of a particular gang.
The symbolic nature of hair also makes it useful as a disguise. When Ponyboy and Johnny must hide on Jay Mountain, they decide to cut or alter their hair in order to push aside their association with the greasers. Hair is ultimately a way of indicating affiliation, personality, and social class.
It isn't so much that they wear greasy hair, but that they put grease in their hair to slick it back. This was a status symbol, as was the long hair that many of them grew. It was for them what Hollister clothing is for many teenagers today - a way to express themselves and also identify themselves. It was also a way to seem more "tuff", as it is put in the book. The slicked back look was in contrast to the more preppy look the Socs were wearing, and it helped the Greasers feel as if they were rebelling against the society that so often put them down.
It sets them apart from other people in society. Also it makes them who they are.
I guess that in some ways, it made them who they are.