Kali is a grudge-bearing pigeon who is aggressive and offensive. In Episode 1 she robbed an old lady and in Series 1, Episode 2 she killed one of her oldest friends as he was became a born-again Christian. She is shown to have a good heart in some circumstances (e.g. showing remorse when she killed the rickshaw driver in Series 2, Episode 2).
Kali does have a murderous side, evidenced in Series 2, Episode 2 when she murdered a rickshaw driver who ran over her meal (a badger, who was alive and did not mind that Kali was eating her).
She used to nest in Amy Winehouse's hair but moved after she drove Kali up the wall.