Bill Kristol is not a neo-conservative. He is a Republican, but certainly not conservative.

Kristol is associated with a number of conservatively aligned think tanks: he was chairman of the New Citizenship Project from 1997 to 2005, he cofounded the neoconservative Project for the New American Century (PNAC) in 1997 with Robert Kagan, he is a member of the board of trustees for the free-market Manhattan Institute for Policy Research, and he is a member of the Policy Advisory Board for the neoconservative Ethics and Public Policy Center. Kristol has also been an attendee at Bilderberg Group conferences.
Kristol was born to Irving Kristol, one of the founders of the neoconservative movement.