Unfortunately, I do not know of any books he personally wrote on the subject (that is not to say he did not write any, just that to my knowledge none are seriously reprinted today). To get a better feel for his background, upbringing, character, and moral and political beliefs I suggest reading some of his letters; many are widely reproduced online. Specifically for the war, it may be interesting to compare his personal letters to his friends and supporters against his professional letters to members of the Continental Congress of the same timeframe - there are some stark differences of tone.

Otherwise pick up a scholarly biography, my personal favorite is His Excellency written by Joseph Ellis. 1776 by David McCullough would be another great choice for a more war centered view, which also follows an abbreviated path of some other founding men and their relationship with Washington. Consider picking up an audiobook of these as they can be long, but great for driving.

As for a true "book" written by Washington that I happen to have on my shelf is Rules of Civility & Decent Behavior in Company and Conversation. This is a pocket dictionary sized book of Washington's self chosen principles of personal conduct and is a very interesting short read that may be enlightening on the specific subject of character you asked about. It is reproduced in its entirety here: http://www.foundationsmag.com/civility.html

Hope that helps, best of luck!