Introducing myself to researchers from my dream school
That is called networking and it is about getting to know people and people getting to know you. It will increase your chances and that's exactly the point.

There is absolutely nothing wrong, immoral, or illegal with it, by the way, it is a necessary skill for all professional careers. Conferences usually have intermissions for that exact purpose, with snacks, refreshments, etc. 

The "standard" way of doing that would be to walk over, politely introduce yourself (don't go all formal), from which school you are and briefly, very briefly, what you are working on. Don't be pushy, don't force the conversation. It is easier if a common acquaintance introduces you (for instance, another researcher that knows you and them). Bonus points if you are familiar with their work/field.

It's easier after they see your presentation, but that's not entirely necessary. The most 'busy' researchers don't usually attend the whole conference, so don't waste chances.

More importantly: forget dream, don't be starstruck. They are just people, it is a school, just like countless others. Something I saw people do wrong: don't get stuck talking shop, especially if you have more time (as in the formal dinner some conferences have). Again, professors/researchers are people, they do more with their lives than just research and certainly have some other interests.