destiny laughing as a young man laughs,

The speaker reiterates the position of his city through the preposition "under." This conveys the struggle of Chicago and presents it as a type of underdog. In spite of that position, Chicago still "laughs," which brings us to another literary device.

The speaker also uses repetition, particularly as the poem closes. Consider the intentional repetition of the word "laughing" in these lines:

Under the smoke, dust all over his mouth, laughing with white teeth,
Under the terrible burden of destiny laughing as a young man laughs,
Laughing even as an ignorant fighter laughs who has never lost a battle,
Bragging and laughing that under his wrist is the pulse, and under his ribs the heart of the people,
Laughing!
Laughing the stormy, husky, brawling laughter of Youth...

In Sandburg's poem "Chicago," the entire first stanza uses personification, as well as an extended metaphor as the writer compares the city to the things people do:

Hog Butcher for the World,

   Tool Maker, Stacker of Wheat,

   Player with Railroads and the Nation's Freight Handler;

   Stormy, husky, brawling,

   City of the Big Shoulders...

Personification gives human characteristics to non-human things. Chicago is a place (a thing) that cannot do human things such as "brawling" or have physical characteristics such as "Big Shoulders."

The extended metaphor makes a comparison that continues throughout several lines. It is defined as:

...a metaphor developed at great length

It lists professions that people have, but assigns those professions to the city, rather than to people. The city, for example, cannot be a hog butcher or toolmaker: only its people can, thereby supplying to the needs of the nation or even the world.

Similes are comparisons of dissimilar things that share similar qualities, using "like" or "as" in the comparison. The poem has several similes:

Fierce as a dog with tongue lapping for action, cunning as a savage pitted against the wilderness...

In these similes, Chicago is compared to dogs and savages. Here are examples of personification, as are the closing lines of the poem. However, these closing lines use brilliant imagery to provide a picture of diverse aspects of Chicago with such clarity, that Sandburg's personification brings alive the spirit of the city, one which he obviously admires. 

In these lines, the words form images in the reader's brain. Sensory details or descriptive details are used—details that appeal to one or more of the senses: in this case, visual senses.

[The five senses] are our primary source of knowledge about the world. Therefore, writing which incorporates vivid, sensory detail is more likely to engage and affect the reader.

To make his writing more engaging, Sandburg uses impressive sensory details: here is the image of a "dusty mouth" contrasted with "white teeth." 

Under the smoke, dust all over his mouth, laughing with white teeth...

Under the terrible burden of destiny laughing as a young man laughs,

Laughing even as an ignorant fighter laughs who has never lost a battle...

Sandburg brings to mind the essence of the city, the life it seems to have—directing the reader's attention to the part of its "body" that refer to the most caring aspects of a human being:

Bragging and laughing that under his wrist is the pulse, and under his ribs the heart of the people,

                   Laughing!

There is joy in the city; it has a pulse that infers it has a life of its own; and it has a heart—but not of its own: it houses "the heart of the people" and an element of the goodness the author feels it is imbued with, found in the word "Laughing!"

Additional Sources:

http://ai.stanford.edu/~csewell/culture/litterms.htm

http://writing.colostate.edu/guides/guide.cfm?guideid=91

"Chicago" is a poem in which the speaker looks at the city of Chicago in two very different ways. On the one hand, the city is seen as a place of prostitution where children are allowed to starve and killers are set free to kill again. On the other hand, the speaker cannot but help be impressed by the hard work and effort that has gone into the making of this city and the industrious nature that characterises it. The literary devices that are used in this poem help to support such understandings. Consider the following example:

Fierce as a dog with tongue lapping for action, cunning
as a savage pitted against the wilderness...

The two similes here both are used to describe the city of Chicago and the kind of people who inhabit it and have made it what it is. The speaker suggests that to get ahead in this kind of world you have to be both "fierce" and "cunning," and his admiration for the workers of Chicago and the city's spirit in this regard is evident.

Note how Sandburg personifies the spirit of Chicago in his picture of a young man who is located:

Under the terrible burden of destiny laughing as a young
man laughs,
Laughing even as an ignorant fighter laughs who has
never lost a battle...

The defiance of the city is expressed in its personification as a "young man" laughing, even though he is "under the terrible burden of destiny." In the face of crushing forces that seem to oppose it the city shows considerable spirit and bravery which make it, for Sandburg, remarkable and worthy of praise.