The American Dream played a huge role during the Founding, it begin when Britain started the Stamp Act, for America it was troublesome. Britain played a role with giving the Americans huge taxation such as, tea, clothes, stamps, etc. America was tired of the whole taxing by Britain, because it ruined them economically. This started the American Revolution, Britain wasn't so pleased of course and immediately took action to this. This relates to the american dream by having the colonies not having their american dream of their own until they got their freedom by throwing the american revolution. Also it relates to the American Dream by Britain taxing the Americans economically leaving the Americans with no choice but to create a revolution.
Sources: http://www.atigun.com/writing/u-s-colonial-consumption/