The American Dream in the post-World War II era, the setting of Miller's play, was essentially the same as it is today: to build a successful life. Financial success and home ownership were certainly a part of it, as were raising a family and living to see one's children graduate from college and succeed on their own. Willy Loman reached for the American Dream, but it eluded him for various reasons.
Despite his years of hard work, Willy's career as a salesman falls apart; at the end of his life, he is unemployed and almost penniless, living on loans from his friend to support his wife Linda. Ironically, as he moves toward old age, the time when he should be enjoying the fruits of his labor, Willy is worth more to his family dead than alive, in financial terms. After he commits suicide, it is the proceeds from his life insurance that pay off the loan on the family's house.
An even more painful failure for Willy is the way his sons turn out. Both Biff and Happy grow up to be irresponsible adults, never succeeding in life in any way, personally or professionally. Willy's dreams of college for his sons end when Biff fails to graduate from high school and Happy does not apply himself in any productive activity.
Willy Loman's American Dream does indeed turn into a nightmare for him, and he himself is responsible for much of it. Part of Willy's tragedy is that he never recognizes how or why his dreams fall apart.
Miller's conception of the American Dream through Wily Loman helps to illuminate the hollowness of a pursuit based solely on material ends. The post war prosperity in America of the 1950s defined "success" as a monetary end, as one where the acquisition of money and the trappings of wealth helped to determine individual worth and how individuals saw one another and themselves. The idea that Wily's dreams are all predicated on wealth and how much money is acquired is the reason why he would actually embracing killing himself for the life insurance money. The lack of a moral or humane component to such a dream might help to explain why Wily's pursuits end up ringing as hollow, his work as a father and husband distant, and why the American Dream as defined by monetary success is futile, at best, and personally destructive at its worst.
The American Dream during the 1950's was to own one's own home, have a good job in a company that one can stay with and expand and gain economic success, have a family, and a family car.
Willy bought into the American Dream and made a good start in the field of sales. However, going over the road to work meant he had little time with his family, lived a lonely life, and as he grew older and economics changed, he fell behind.
Willy actually was living the American Dream in his youth. His boys had a home they lived in and his wife was a home mother taking care of her children. He may have never owned the mortgage of his home, but many Americans grow older and do not pay off their home.
I can not say that he did not have the American Dream because it is a concept not a solid thing. In his death he left the mortgage to his wife. His life became a nightmare for him, but he always seemed to be living in a dream world.
Usually the man brought the money in the household and usually involved things like accounting or being what Willy Loman was, a salesman. He had a wife and kids and although this represented his dream, he was in debt of the house and he could not work properly. His mental state was in complete disarray and his version of "success" was the cause of the family's jumbled end. His flawed view on what classifies as success and his attempt to live it forced him to live in a fake view of the world and himself, and this was a part of his downfall.