As you can see from responses you've already gotten, asking questions here isn't necessarily going to give you answers you'll find helpful. I expect that part of it is that it's a lot harder to get your point across accurately in writing than it is face to face, especially when you've only made your opening statement and there hasn't been a chance to clarify any confusion that people have. If you'd said you wanted to go to a Ford assembly plant and talk to all of the black workers, I expect that some people would figure that you thought all the workers were black and others would figure that's the demographic you wanted to focus on. Similarly, some would assume your motives were racist and others would figure there was some other reason you wanted to focus on.Maybe a few would try and answer your questions.
Other than 100% of the people here having internet access, I don't imagine there's much difference between the people here and the lottery players you'll find in stores, assuming you go to a variety of areas. With that in mind, if I was in your shoes I'd just wade in and get started. To get the information you need you'll have to ask a lot of questions, so it should be easy enough to start out with innocuous questions and let things develop from there. Unless you just want to take a poll you shouldn't want to ask everyone the same questions in the same order anyway.
I expect that one of the things you'll find is that different people have different ideas about what the "American dream" is in the first place, and the destination you start with may not be where you end up. Personally, I don't see why anyone who thinks they've already got their dream would play the lottery (other than a gambling problem or some other issue), so I don't see how they'd fit in a documentary abut instant riches and the American dream. Clearly there are other people for whom the lottery really is the only (though very unlikely) way of acheiving their (financial) dreams. I'm guessing, though I'm not necessarily convinced, that most players fall somewhere in the middle.
If it helps any, here's my personal take on the lottery and the dream. I'm pretty sure that when I'm on my deathbed I won't be laying there wishing I'd worked harder. If I wanted to spend more time busting my ass and not having the time to enjoy what that brings me I could have more money and be more successful the way some people define success. A bigger house might be nice, but no amount of money will buy a house with the kinds of thinsg I'm most interested in. The world is full of all sorts of fun things, and there's never enough time to do everything you'd like to do. Some require money, and they all require time. I know my chances of winning the lottery are extremely small, but it is my only chance of having the kind of money that would let me do about anything I could realistically want to do and have the time to try and do most of it. Just because I see my cup as half full doesn't mean I don't know it only has half of what it could hold.