I think that Hamilton wanted to prove he was the right choice when Washington picked him, and wanted to prove that he was more than just an illegitimate orphan. I also wonder if Washington influenced Hamilton or if Hamilton influenced Washington? Washington says, "Alex, listen. There's only one way for us to win this Provoke outrage, outright," but before that Hamilton had been saying that Washington was despondent, and he was writing all of his correspondents. I think that the thought of failure on both of their parts influenced them both to step up and take charge. Washington started dishing out orders once he found out that the calvary wasn't coming, and Hamilton backed him up the whole way, until he promoted someone else over him.
Hamilton follows Washington's orders and does exactly as he says, and expects to be promoted, but instead Charles Lee is promoted. Which turns out to be the wrong decision, because Lee is a big scaredy cat and won't step up and take charge and go for it. When Lee fails, he begins to trash talk Washington, and Hamilton isn't very happy about that but Washington says, "Don't do a thing. History will prove him wrong. We have a war to fight, let's move along." I think that Washington knew that Hamilton only wanted the best for him, but knew that if he went after Lee, that he would blow his shot so he dismissed the bad talk and moved himself, Hamilton and the war right along in order to accomplish the ultimate goal and win.
work cited: