How are Hally's hopes depicted in the play?

Hope is an interesting concept and/or theme for this play. In general, I don't think this play is all that hopeful of a play; however, there are hopeful moments in the play. One such instance occurs right when Hally enters the scene. The stage directions tell readers that Sam begins doing the quickstep, and that he is a very accomplished dancer. Hally enters and applauds. He is very excited about Sam's actions, and it turns out that Sam has an upcoming competition. Both Sam and Hally seem excited and hopeful about Sam's chances in the competition.

HALLY: Think you stand a chance?
SAM: Let's just say I'm ready to go out there and dance.
HALLY: It looked like it. What about you, Willie?

This is an important, small exchange between Hally and Sam because it shows that Hally is capable of caring for somebody other than himself. By the end of the play, that isn't the case. Hally is so careless about anybody other than himself that he starts throwing around racial comments and demanding to be called Master Harold.
Shortly after the dancing part, readers get a very different kind of hope from Hally. The weather is bad, and he says that means he gets a quiet afternoon.

HALLY: [Gosh], what a lousy bloody day. It's coming down cats and dogs out there. Bad for business, chaps . . . [Conspiratorial whisper.] . . . but it also means we're in for a nice quiet afternoon.
SAM: You can speak loud. Your Mom's not here.

His mom isn't home either, and just for a moment readers believe that he is excited for some time alone at home with no parents. It's just Hally and his boys; however, Hally's hope becomes a bit tainted when readers realize that he is hopeful that his dad won't be coming home for a long time.

HALLY: [With conviction.] No! It can't be. They said he needed at least another three weeks of treatment. Sam's definitely made a mistake.

Hally's hope is that his dad is gone for a long time. This seems like a rather morbid hope, but it turns out that Hally's father is a terrible and abusive father. Hally genuinely hopes for minimal interaction with his father.
Despite Hally's warped relationship with his father, Hally remains an overall happy and upbeat kid around Sam. That changes when it is confirmed that his father will be coming home; however, Hally does show readers that he has a hopeful outlook on the world itself. This is shown to readers as Hally and Sam banter, tease each other, and study. At one point, Sam tells Hally about a physical form of punishment that still occurs. Sam doesn't hold back in his description, and Hally asks him to stop. Sam says it's the truth, and Hally admits that he hopes it won't always be that way.

HALLY: I've heard enough, Sam! . . . It's a bloody awful world when you come to think of it. People can be real cruel.
SAM: That's the way it is, Hally.
HALLY: It doesn't have to be that way. There is something called progress, you know. We don't exactly burn people at the stake anymore.

Sam is skeptical, and he forces Hally to explain a bit more. It's here that Hally openly admits to his hope in social reformers.

HALLY: [A world-weary sigh.] I know, I know! I oscillate between hope and despair for this world as well, Sam. But things will change, you wait and see. One day somebody is going to get up and give history a kick up the backside and get it going again.
SAM: Like who?
HALLY: [After thought.] They're called social reformers. Every age, Sam, has got its social reformer. My history book is full of them.