As with so many things in the Revolution, the answer will change depending on when in the war you're asking about. Early in the war, many soldiers elected their own company or regimental officers; this was, after all, a Revolution about representation and the consent of the governed. As the national and state military hierarchies developed, however, those in charge quickly moved to undo this system. While soldiers who elected their officers often picked men with the necessary qualities for leadership - the abilities to read, write, inspire, and pay their men - this was not universally true. Elected officers were also forever beholden to their men, and thus Washington, Congress, and the state governors feared they would serve the private soldiers' selfish and short-sighted interests over the best interest of the cause, especially when units were ordered to perform difficult, dangerous, boring, or otherwise distasteful duty. Creating an environment where private soldiers thought they got to run, or even have a meaningful say, in army operations was a nightmare for military discipline as the elite viewed it at the time.

Congress and the state governors quickly worked out a system where they would control commissions in the Continental Army, while state governments handled militia commissions. At the top level, Congress appointed Washington commander-in-chief, along with six major generals to assist him and run sub-commands of the army. All of these men were politically well-connected, and all had prior military experience. As the war dragged on, Washington was able to get most of his favorites - Knox, Greene, Lafayette, von Stueben, Wayne, and others - into these positions.

At the state level, governments selected gentlemen - that is, men of wealth and community standing - to serve as officers. Greene is actually a good example of this. He was successful businessman and active Patriot in the pre-war period, and played an active role in raising soldiers for the rebel siege of Boston. However, his political opponents alleged that his slight limp barred him from military service, and he was disqualified from the election for officers. In protest, he served in the ranks until Congress took over the duty of assigning commanders, and Greene promptly received the biggest promotion in American history, bouncing from private all the way up to brigadier general.

Having Congress control appointments to the army was something of a nightmare for Washington in the early-to-middle days of the war, with Congress offering high commissions to seemingly ever weirdo foreign adventurer that washed onto American shores. This stopped after 1778: Washington's political annihilation of Charles Lee cowed Congress into deference to him on military matters, and weirdo foreign adventurers could just join the French army instead.

Merit played a greater role in promotion in the Continental Army than it did in the British and ESPECIALLY the French armies of the period - the latter required patents of nobility as a prerequisite to receiving an officer commission. Still, seniority was a key component of both promotion and untangling the social pecking order for the ever-fussy officer class. Meritorious service was the best way to jump from enlist to NCO to officer, but even Washington did not have the power to arbitrarily advance someone in rank.

Sources: Martin and Lender, A Respectable Army

Royster, A Revolutionary People at War.