Could a pair of binary black holes form within a star?
I doubt two black holes could form within a single star, but they might form from a contact binary. That's a pair of stars so close that they touch and share a single atmosphere. For the stars that made the black holes for the LIGO gravitational wave discovery, just consider two very massive stars that are touching or nearly touching. They would share a single gaseous envelope for an atmosphere. Each one separately formed a black hole when it died, and then their two black holes merged.