Michael and Jana Novaks have an excellent article in The American Enterprise, "Washington's Faith and the Birth of America".
Almost everything about George Washington was hard-earned, and his faith was no exception. Although he ended up owning a library of nearly 1,000 books, some 40 or so concerning religious questions, his preferred teacher was experience. In his father’s line there was at least one Anglican cleric, and his mother was unusually devout and quite attentive to the religious life of her children. But Washington’s faith mostly grew out of his diligent efforts at self-improvement.
Washington studied the thinking of British generals and European monarchs, the manners of Indian chiefs, the habits of fur trappers of the frontier, good farmers and bad, and trustworthy and untrustworthy merchants. He pondered the ways of Congress and the surges of public sentiment. He watched closely the inner passions of his own fighting men.
He learned how all sorts of humans reason, what they fear, what attracts them, and what moves them to action. Washington was nearly a genius in getting the best out of his own hotly rivalrous cabinet. He had enormous common sense and a wordless instinct for how things actually work in the real world. He understood the dramatics of gesture, raiment, and dress; he understood the role of imagination as well as reason in life, of passion as well as logic. His was not a highly verbal nor academic mind. But his practical judgments were sure.
Washington knew that the tasks he undertook were too big for him, that he was too unlearned and lacked some of the necessary gifts, that the odds he faced were steep. He truly, genuinely, feared failure.
He got nearly his fill of excruciating disappointment on many occasions. At Monongahela, on Long Island, at Valley Forge, on the icy Delaware, at the Newburgh meeting, he could taste failure, it came so close. His was not a fake humility. He knew he had many limitations, and that given the immense tasks handed to him, he might embarrass all who depended on his success.
So how did George Washington persevere? As he acknowledged many times, he trusted “Providence.” George Washington had a silent ally to whom he regularly gave thanks, publicly and privately.
There are excellent observations and insights on the historical context of Washington's beliefs; one key fact is that 18th century Anglicanism was uneasy about religious "enthusiasm", having swung back and forth between formalism and Puritanism for a century and even then deciding what to think of George Whitefield and John Wesley's Methodist movement. A very self-restrained man on matters of the heart, Washington simply couldn't be expected to display the outgoing expressions of faith encouraged by the modern evangelical church.
Some historians seem to fear religious interpretations of Washington. More recent biographers often suggest Washington was at best a lukewarm Anglican, that he was more a deist than a Christian, and that his concept of “Providence” was closer to the Greek or Roman “Fate” or “Fortuna” than to the Biblical God. Yet Washington’s own stepgranddaughter, “Nelly” Custis, thought his words and actions were so plain and obvious that she could not understand how anybody failed to see that he had always lived as a serious Christian. As she wrote to one of Washington’s early biographers:
It was his custom to retire to his library at nine or ten o’clock, where he remained an hour before he went to his chamber. He always rose before the sun, and remained in his library until called to breakfast. I never witnessed his private devotions, I never inquired about them. I should have thought it the greatest heresy to doubt his firm belief in Christianity. His life, his writings, prove that he was a Christian. He was not one of those who act or pray “that they may be seen of men.” He communed with his God in secret.
The article is adapted from the Novaks' book Washington's God. I also recommend John Eidsmoe's Christianity and the Constitution for more discussion of the deism question with regard to the Founders (Eidsmoe debunks most of it, btw).
Well worth reading here: http://www.taemag.com/issues/articleID.19111/article_detail.asp