How does the song "Big Time" by Peter Gabriel relate to The Great Gatsby's theme of the American Dream?

The lyrics for this song reflect the common understanding of the "American Dream": amass wealth at any price in order to gain the respect of others and all the creature comforts money can buy.

The opening lines seem like something Gatsby himself would sing:

"I'm on my way I'm making it, huh!
I've got to make it show yeah, hey!
So much larger than life."

Think of how Gatsby was so determined to gain wealth in order to show it to Daisy and prove he was worthy of her love. 

The second stanza addresses the idea of poor, small towns being somehow inferior to the big city and wealth. Having a lot of money means you must be smart. Poor people must not have the brains to be able to achieve wealth and status. Compare this to the way Gatsby created an elaborate lie about his past in order to conceal the fact he was from a poor family in a small town:

"The place where I come from is a small town
They think so small, they use small words
But not me, I'm smarter than that,
I worked it out..."

The lyrics also refer to owning a lot of material possessions, which is often an aspect of the American Dream and Gatsby's, Tom's, and Daisy's life:

"I've had enough, I'm getting out
to the city, the big big city
I'll be a big noise with all the big boys, so much stuff I will own..."

But the lyrics don't actually praise this attitude, just as Fitzgerald doesn't esteem the excesses of the Roaring 20's:

"My parties have all the big names and I greet them with the widest smile. Tell them how my life is one big adventure and always they're amazed when I show them 'round my house to my bed I had it made like a mountain range with a snow white pillow for my big fat head."

Although the beginning of this verse brings to mind Gatsby showing Daisy and Nick around his house, the last line about "my big fat head" is actually insulting the speaker. Someone with a 'fat head' is self-centered and vain. That doesn't apply to Gatsby, but it certainly could apply to Tom or even to Daisy.

By the end of the song, the speaker is out of control with the desire for 'big' everything: his car gets bigger, his house gets bigger, his eyes get bigger, his mouth gets bigger, his stomach gets bigger. The idea of eyes getting bigger refers to constantly wanting more and not being satisfied; a person with a big mouth is usually obnoxious; a big belly shows the physical deterioration of his body because of his excesses.

So the lyrics relate to the American Dream because they refer to the overwhelming desire to gain money and power, but they also show the inherent shallowness and discontent that may accompany that dream.