LONDON—In a chic Kensington house near Harrods department store, the former head of the Formula One racing circuit has set up a small business with a big mission: to extract payback from News Corp. Chairman and Chief Executive Rupert Murdoch.

The house serves as an office for Max Mosley, who in 2008 became the object of international ridicule when News Corp.'s News of the World weekly tabloid published details of a sadomasochistic romp he'd had with five women. The paper reported the orgy had a Nazi theme, an especially sensitive...