I dont think rin told him to kill her coz she was so shocked by kakashi's action. i think:
(1) rin was being controlled by enemies the whole time to spy on konoha with/without her consciousness.
-and there probably no other way out to let her free from being controlled.
-or kakashi had to kill her because if he didn't,rin would probably go back to enemies and leak all the
secrets of konoha with/without her consciousness. you can say,more like Kisame's situation (before he
joined Akatsuki).
(2) Rin was really on enemies' side. But at that time,no body thought Kakashi had figured it out. So that's
probably why Rin was shocked when Kakashi killed her,and the enemies were all like "he got us!".
(3) They obviously wanted Rin's body to retrieve information from her,just like what Inoichi did to the other
body of Pein.