The "separation of church and state" is one of the most misunderstood parts of the U.S. Constitution. The relevant text is found in the first sentence of the First Amendment, and it states:
"Congress shall make no law respecting an establishment of religion, nor prohibiting the free exercise thereof... "
The first part of the sentence is the Establishment clause, and the second part is the Free exercise clause.
All it means is that the government may not set up an official state religion or pass any laws that would have the same effect. Likewise, it also may not pass laws that prevent you from expressing your own religious devotion in whatever manner you choose.
The First Amendment requires the government to be neutral with regard to religion, and further requires that neutrality to be inclusive of religious faiths, not exclusive of them. In fact, the government is required to consider religious faith when it acts, weighing how to accomplish its goals in a manner that restricts religious expression to the minimum extent possible.
Now, on to Jeff Sessions' comments...
There are two Jeff Sessions at play here. One of them is Jeff Sessions, Attorney General of the United States, and the other is Jeff Sessions, a man from Alabama.
AG Jeff Sessions is required by oath to uphold the Constitution in his official capacity as Attorney General, including the Establishment and Free exercise clauses. Citizen Jeff Sessions is just an ordinary guy who is free to express whatever religious beliefs he wants.
As a man, Jeff Sessions is an evangelical Christian from a heavily evangelical state (they don't call it the Bible Belt for nothing). His religious faith informs and guides his thinking and worldview as Attorney General. Sometimes his faith and his job are in agreement with each other. Sometimes they are not. But in this particular case they are, and this is what he was saying.
It is not the policy of the United States government to use the Bible or any other religious text as the basis of law. However, sometimes the law aligns with religious teaching, and that's perfectly understandable, since the law and religion both ultimately serve (or at least attempt to serve) a moral purpose.
You can separate church and state all you want, but you cannot separate religion from politics – especially in a country like the US where religion is a much bigger part of the daily lives of its citizenry than for other western democracies.