The song opens with the phrase "I am not throwing away my shot", which is also one of Hamilton's catchphrases. It symbolizes his ambitious nature to not waste any opportunities and use any means to climb up the ranks. This phrase is repeated many, many times in the song, and also foreshadows Hamilton's duel with Aaron Burr (more on this later....).
The context of the song is when Hamilton first gets to New York, and is looking for something to be a part of, specifically the revolution. This song highlights his naïve and arrogant nature with lines like "Try'in to reach my goal, power of speech impeachable" and "Only nineteen, but my mind is older".
He then expresses his discontent with Britain through the lines "Essentially, they tax us relentlessly". As mentioned briefly in the introduction, Britain imposed a series of taxes on the colonists without consulting or passing it by any political leader in the colonies. This led to the protest motto "No taxation without representation". This was developed when most colonists still didn't want to be independent from Britain, but as we can see from the song, Hamilton predicts a revolution and dreams of "a colony that runs independently".
Next in the song, we learn the background and beliefs of Hamilton/Burr's friends. Lafayette wants to live in a country without a monarchy (so the opposite of what Britain had). He keeps this belief when he goes to France after America becomes independent and contributes greatly to the French Revolution. Mulligan is a tailor's apprentice, which allows him to meet British soldiers frequently, hence gaining their trust. He later exploits this trust to act as a spy for the Revolutionary Army. Laurens was an abolitionist, and he always dreamed of freeing slaves. During the war, he raised a brigand of 3,000 black soldiers and promised them freedom in return for fighting.
Burr then comes in and brings the group back to reality. He warns them again of the risk of speaking for the revolution, but Hamilton doesn't listen.
Finally, in the last part of the song Hamilton seems to have solidified his purpose in life. He never really thought about his future, which he expressed in the lines "See, I never thought I'd live past twenty" and "for the first time I'm thinkin' past tomorrow".
His part in the end also brings into picture the current situation of America. The country is at a point where its people doesn't know whether to become independent or not, and doesn't know what would happen if they do become independent. This is best summarized by the line "Are we a nation of states? What's the state of our nation?"