As Hillary Clinton was beginning her job as President Obama’s chief diplomat, federal agents observed as multiple arms of Vladimir Putin’s machine unleashed an influence campaign designed to win access to the new secretary of State, her husband Bill Clinton and members of their inner circle, according to interviews and once-sealed FBI records. Some of the activities FBI agents gathered evidence about in 2009 and 2010 were covert and illegal. A female Russian spy posing as an American accountant, for instance, used a false identity to burrow her way into the employ of a major Democratic donor in hopes of...