At age 4, Naruto swore he would become Hokage. At age 7, he abandoned that dream, but he doesn't tell anyone that. He shouts about it, at any chance he has.
He does it because it gives Hinata strength, makes her feel good. When Hinata watches Naruto, she is filled with confidence; the belief that she isn't worthless, and that she too, can become strong. Naruto continues the façade because it gives Hinata hope.
When Konohamaru watches Naruto, he is inspired to train harder; he throws himself at his lessons. Naruto keeps the mask up, because the 'rivalry' spurs Konohamaru to train even harder; Naruto has grown strong, and that gives Konohamaru hope, that he can become like Naruto.
He reinforces his dream, because it reminds Neji. Reminds him that the chains of fate can be broken; that a caged bird, with time, can learn to unlock its prison gates and fly free; fly with nature. He shouts about being Hokage, because it gives Neji hope.
Naruto continues to proclaim his dream, as reminder. Tsunade became the Hokage because of his belief, and he's not going to stop anytime soon. Tsunade sees her little brother, Nawaki, and her lover, Dan, in him. Naruto doesn't want to extinguish her fragile hope that Naruto won't fall; that he won't succumb to the curse.
Naruto may no longer have hope, but the people he considers precious do, because Naruto pretends he has hope. And without hope, what reason is there in living?
