This won't be much help, but it tells you some about his lack of formal education: http://www.nps.gov/archive/fone/clas...on_student.htm
As you said, he was a self-made man, but he had no formal education beyond age 16. He obviously learned a great deal, much of it from his older brothers and much from reading and experience, especially as a surveyor.
This being the case, you have a difficult project and may well find something in his own writings. He said little about religion anywhere and apparently had none. There is an interesting blog here: Washington's religion
Protestant religion would tend to show at least an indirect influence from Plato, but as you can see, there is little there.
In the Continental Congress he famously remained silent. He wore a military uniform and accepted the office of Commander-in-Chief. More importantly, he learned much from those about him, and they were awash with philosophers like Locke and Montesquieu especially and with the other enlightenment philosophers. Some of that had to rub off on him. Maybe you can find some direct evidence in his writings. He was a whig, and that in itself suggests indirect influence from Locke.
So you will need to spend some time time looking for indirect influences in his own writings. He was not accustomed to quote people, but to state his case plainly and briefly.
One person who did have some influence on him was Martha Dandridge Custis. Another was Thomas Paine.
I hope that gives you some ideas where to start, but it's going to be a tough one to do.