Her Resonse is: Response: After a Unique [stark] character is killed, discard Arya Stark from play (cannot be saved) to choose and kill a character with STR 3 or less.

Can I use her ability when iam killing her, and then trigger her ability when she is moribound to Discard Arya Stark from play to kill a character ???

No, there is a rule in FAQ document that forbids specifically to make moribund a card that is already moribund.