In the late sixties,
a rumor that the Beatle's legend, Paul McCartney, died and was replaced by a look-alike began to circulate among fans..
A) a rumor that the Beatle's legend Paul McCartney died and was replaced by a look-alike began to circulate among fans.( Simple past does not convey the intended period of action)
B) a rumor that the Beatle's legend Paul McCartney had died and was replaced by a look-alike began to circulate among fans.( had died is correct,simple past shows a gap in replacement)
C) a rumor that the Beatle's legend Paul McCartney had died and had been replaced by a look-alike began to circulate among fans.( no delay in action,correct choice)
D) a rumor began to circulate among fans that the Beatle's legend Paul McCartney had died and had been replaced by a look-alike( that incorrectly modifies fans)
E) a rumor began to circulate among fans that the Beatle's legend Paul McCartney died and was replaced by a look-alike.( same as in D)
Will go with choice C,though B is also tempting.