Although I'm not an expert on the American Revolutionary War, I am an economic historian and have extensively studied Alexander Hamilton's absolutely pivotal role in the creation of the modern american political and economic system.

Hamilton's close personal relationship with Washington and his role as Secretary of the Treasury (the government's money guy) led several scholars to call him Washington's "Prime Minister." In spite of opposition by the likes of influential figures like Thomas Jefferson, Hamilton managed to construct a large, powerful Federal government to the detriment of individual states, with a consolidated national debt structure and a standing army. John L. Harper in his "The American Machiavelli," concludes that Hamilton not only was a central, indispensable figure in the construction the apparatus of the American government, but also lay the foundation for the underlying philosophies which would guide American foreign policy: opportunistic but pragmatic, both imperialist and isolationist, "strength through peace and peace through strength."

I'm sorry if this sounds vague, but Hamilton's exploits has filled entire volumes, like Harper's book cited above. Is there anything specific you're interested in?

Jumping off /u/AlviseFalier's excellent answer about Hamilton's post-war career, the biggest contribution Hamilton made during the war was politically destroying Washington's rivals for power. Though revered as the Father of His Country by some even during the war, Washington has a miserable performance record for much of the first three years of the war. While he did force the British to evacuate Boston, Washington ended 1776 by losing a series of battles around New York City and the entire state of New Jersey for a few weeks. The following year, he lost the capital city of Philadelphia to the enemy. Washington owed his continued command to his lightning strikes on Trenton and Princeton in December/January 1776/7 (and the fact that having a Virginian in charge of New England troops cemented the Revolution as a national struggle).

Washington's spotty record made him vulnerable in the eyes of two of his Major Generals: Charles Lee and and Horatio Gates. Lee was (despite the verse he gets in Hamilton) the most experienced soldier in America, having fought in the British and Russian armies in the decades before the Revolution. Gates was a former British Army officer and the winner of the Revolution's largest and most important victory, the Battle of Saratoga. Both men curried favor in Congress by telling Congressmen like John Adams what they wanted to hear: the Patriots could win the war with a cheap and non-threatening army of militiamen, if only they were competently led. Coming into 1778, Washington's continued employment as Commander-in-Chief was as doubtful as it was at any point during the war.

Washington, to his credit, did not personally engage in the bitchy sniping and gossiping of army politics. He did, however, surround himself with a cadre of absolute vipers, men who liked Washington personally and, in the case of his chief aide-de-camp Hamilton and chief of the artillery Henry Knox, owed their careers to Washington. Hamilton was the ringleader of efforts to push back against these would-be replacements.

Hamilton's biggest coup came after the Battle of Monmouth in June, 1778. Lee commanded the American vanguard, but somewhat bungled his attack on the rear of the British army. After several hours of fighting, Lee began a more-or-less orderly retreat to higher, more defensible ground. In the midst of this retreat, Washington arrived with the rest of the army. He assumed command, ordered American troops to hold more or less where they were, and sent Lee to organize the American rear. Late in the battle, Washington and Lee worked together to stabilize the shaky American line, and Washington thanked Lee for his good service.

Hamilton saw an opportunity to seize on Lee's seeming under-performance early in the battle, and began writing anonymous letters to Congressmen and newspapers exaggerating the severity of Lee's repulse and ramping up the anger Washington supposedly showed Lee when he relieved him of command (a FAR harsher reading of "assigned him to the rear" than it perhaps warranted). Lee, always thin-skinned and temperamental, let this rumor campaign get under his skin, and he began to publicly accuse Washington of twisting events to destroy his career. Paranoid ravings against a respected superior is not a good look on anybody. By the time Lee demanded the matter be investigated in a court martial, he had become politically toxic, and the court martial basically confirmed Hamilton's version of events. The court relieved Lee of command for a year, sending him into a tailspin where his conduct became increasingly reckless and disruptive. After losing a duel to Washington's aide John Laurens, Congress fired Lee.

Gates, seeing what happened to Lee, made it a point to publicly and privately affirm his loyalty to Washington. Still, Hamilton made it a point to take every opportunity to wound Gates politically. He made sure everyone knew that Gates fled the field at Camden as his army's lines collapsed, and later may have used one of Gates' aides as a patsy in a plot to have the army's officers ALMOST mutiny.