What is kennedy referring to when he says he'' had a member of my family killed

This was actually part of a longer portion of his speech to reporters in 1968 when Martin Luther King got killed. The complete version of this portion is, "For those of you who are black and are tempted to be filled with hatred and mistrust of the injustice of such an act, against all white people, I would only say that I can also feel in my own heart the same kind of feeling. I had a member of my family killed, but he was killed by a white man." 

With this, he was trying to reach out to the black people not to hate white people and have a stereotypical view of white people from a negative perspective. He states that he had a family member killed as well by a white man.

If you were talking about Robert F. Kennedy's speech in Indianapolis after MLK was shot and killed, i believe he was talking about how he was close to him and considered him as a brother, This i believe was said so that way young black Americans would not live in anger of fellow white Americans since a white american was behind the assassination of Martin Luther King Jr.

imparadis't in one anothers arms  

the happier eden, shall enjoy their fill  

of bliss on bliss, while i to hell am thrust,  

where neither joy nor love, but fierce desire,  

among our other torments not the least,  

still unfulfill'd with pain of longing pines;  

yet let me not forget what i have gain'd  

from their own mouths; all is not theirs it seems:  

one fatal tree there stands of knowledge call'd,  

forbidden them to taste: knowledge forbidd'n?  

suspicious, reasonless. why should their lord  

envie them that? can it be sin to know,