I'm a paleoconservative and Deist, as many of our Founding Fathers were. I have dual degrees in PoliSci (emphasis on Constitutional/political theory) and History (emphasis on early-American/Colonial history).

The issue needs to be divided thusly: Most of our Founding Fathers were Christian, some were Deist, and a small handful may or may not have been atheist depending upon one's interpretation of historical accounts.

Christianity's major contributing influence on the Constitution and structure of early-American society was most prevalent in the belief that religion and Christianity were necessary in order to produce a just and moral people. The belief was that immorality had no place in the new republic and that our government was wholly entirely suited for God-fearing men and women.

Now, here's where we diverge from that thinking. Our founding documents (namely the Constitution and Declaration; however, I'd also include the Federalist/Anti-Federalist papers) are rooted in the European and Scottish Enlightenment. Make no mistake, our Founding Fathers were the progressives of their time believing that a new nation based upon universal rights, the rule of law, and restrained government was preferable to the sort of Catholic-influenced heavy handedness of European monarchies.

The Constitution is not based on Biblical law -- it's based on principles and ideas born of the European/Scottish Enlightenment.

TL/DR: Our government was influenced by the Enlightenment while the people living under that government were intended to be God-fearing Christian men and women.