This is a great question and Miller's work, Death of a Salesman is directly related to the the American Dream in an inverse way. In view of this, it is best to define the American dream first and then show how the work addresses this.
The American dream is an ideology that says that freedom and hard work will lead to prosperity. So, anyone who has a good work ethic will make it in America. America is the land of great opportunities. The beauty of this dream is also that it is apart from a person's upbringing, nationality, or anything else. All that matters is hard work and the freedom that the United States provides.
Miller's Death of a Salesman is the dark side of the American dream. Willy is living under its dream without ever being successful. This kills him in the end as he commits suicide. His lack of success and most likely the lack of success of his children is not something that he can handle. This is why throughout the work, we hear him mumbling and living in a dream world. He has created a tragic world for himself.
So, we can say that Willy believes in this American Dream too much and this is his problem. His inability to achieve success defines his life and worth. In the end, death is the only solution he can see.
As was mentioned in the previous post, the American dream is the belief that through hard work and dedication individuals can attain financial success, which results in living a fulfilled, content life. Throughout the play, several characters attain the American dream, while Willy and his sons miserably fail to reach their dreams of living a financially secure, fulfilled life. Through Willy Loman's character, Miller examines the vain pursuit of attaining the American dream. Willy Loman becomes obsessed with his version of the American dream after witnessing his brother's financial success. However, Willy believes that he and his sons can attain the American dream by simply being well-liked and popular. Willy's expectations for Biff and Happy become unreachable due to the wrong guidance and advice he gave them as children. Willy becomes so obsessed with the American dream that he damages his relationship with his sons and eventually commits suicide in order for his family to receive life insurance money. Miller essentially explores what happens when individuals become obsessed with the American dream and reject other aspects of their lives in order to attain it.