First, we should note that it is always dangerous to talk about the "character" of a people as such discussions will always be overgeneralizations that are not true to the way many people thought.
That said, the American character at the time of the Revolution had moved away from strict Puritan piety and more towards an Enlightenment set of values. These values and interests can easily be seen in Franklin's beliefs and interests.
For example, Franklin had broken away from the dogmas of his Puritan parents. Franklin himself was much more of a deist. He preferred to think about God in a more logical and rational way than the Puritans had. This can be seen in the fact that he created his own book of prayers (much as Thomas Jefferson created his own bible) that was drawn both from religious and from scientific sources.
Franklin's interest in science was very much an Enlightenment trait as well. This can be seen in his electrical experiments as well as in his practical inventions.
In general, the American character was coming to center around Enlightenment values and interests. This change can clearly be seen in Benjamin Franklin.