At the time of the signing the US consisted of 13 colonies under the rule of England's King George III. Leading up to the signing, there had been growing unrest in the colonies surrounding the taxes that colonists were required to pay to England. The major objection was "Taxation without Representation" -- the colonists had no say in the decisions of English Parliament.
Rather than negotiating, King George sent extra troops to the colonies to help control any rebellion that might be arising. The following timeline will give you a crash course in the history that lead to the signing of the Declaration of Independence and America's break from British rule.

1774 - The 13 colonies send delegates to Philadelphia, Pennsylvania to form the First Continental Congress. While unrest was brewing, the colonies were far from ready to declare war.

April 1775 - King George's troops advance on Concord, Massachusetts, prompting Paul Revere's midnight ride that sounded the alarm "The British are coming, the British are coming." 
The subsequent battle of Concord, famous for being the "shot heard round the world," would mark the unofficial beginning of the American Revolution. 

May 1776 - After nearly a year of trying to work our their differences with England, the colonies again send delegates to the Second Continental Congress. 

June 1776 - Admitting that their efforts were hopeless, a committee was formed to compose the formal Declaration of Independence. Headed by Thomas Jefferson, the committee also included John Adams, Benjamin Franklin, Philip Livingston and Roger Sherman. 

June 28, 1776 - Jefferson presents the first draft of the declaration to congress. 

July 4, 1776 - After various changes to Jefferson's original draft, a vote was taken late in the afternoon of July 4th. Of the 13 colonies, 9 voted in favor of the Declaration; 2, Pennsylvania and South Carolina voted No; Delaware was undecided and New York abstained.
John Hancock, President of the Continental Congress, was the first to sign the Declaration of Independence. It is said that he signed his name "with a great flourish" so "King George can read that without spectacles!" 

July 6, 1776 - The Pennsylvania Evening Post is the first newspaper to print the Declaration of Independence. 

July 8, 1776 - The first public reading of the declaration takes place in Philadelphia's Independence Square. The bell in Independence Hall, then known as the "Province Bell" would later be renamed the "Liberty Bell" after its inscription - "Proclaim Liberty Throughout All the Land Unto All the Inhabitants Thereof." 

August 1776 - The task begun on July 4, the signing of the Declaration of Independence, was not actually completed until August. Nonetheless, the 4th of July has been accepted as the official anniversary of United States independence from Britain. 

July 4, 1777 - The first Independence Day celebration takes place. It's interesting to speculate what those first 4th festivities were like. By the early 1800s the traditions of parades, picnics, and fireworks were firmly established as part of American Independence Day culture.