The most famous person to have passed himself off as an Indian is Asa Earl Carter, who called himself "Forrest Carter" when passing himself off as part-Cherokee. He is most famous for having written the allegedly autobiographical novel The Education of Little Tree.
It is hard to know what Carter's motives were. In his earlier life, he had been a white supremacist. Among other things, he founded the North Alabama White Citizens Council, which was one of many "white citizens' councils" that sprang up in the wake of the Supreme Court decision in Brown v. Board of Education. Carter is also sometimes credited with writing George Wallace's famous line "segregation today, segregation tomorrow, segregation forever."
It is not clear why a white supremacist would try to pass himself off as a Native American. It could have been for artistic purposes, but he assumed this identity long before he wrote Little Tree and so the connection is not obvious.