I am no doctor but I know it isn't psychopathy. Psychopaths exhibit a loss of empathy and emotions due to anti social personality disorder. They usually are very calculating and don't show signs of remorse. She still feels empathy. She believes that walkers are still alive. That's why she tried to smother Judith. She thought that Judith wouldn't really die and she would end up being quiet. That's why she killed her sister. She didn't believe that she would really die. That's why when Carol killed her "friend" she freaked. She believed that once you die as a zombie, you really die. Because she was so young when the outbreak happened, I think that her developing mind got confused with the whole life and death concept because zombie resurrection was thrown into the mix.
Also unlike a psychopath, she clearly displays emotions. True she does try to make people believe that she's scared of zombies when she's around other people, but she definitely showed regret and panic. She regretted disappointing Carol. When she panicked she had to calm down by looking at flowers.
Lizzie isn't the only one though. It seems to be a recurring theme on Walking Dead that there's someone having trouble accepting that the walkers aren't "people" anymore and are dead. We saw it a little in Hershel and the Governor. Each of them may have responded differently but they all at one point found it difficult to understand life and death. Perhaps it's also bringing into question what separates them from people other than the change. They were human and alive once. People can be monsters like walkers too. What is it that separates them from us? They are trying to survive just like we are and are killing people like some people do. The answer is one: a heart beat, two: rationality, three and most importantly in this show: emotion and empathy. I think that they were really trying to stress that towards the end when they killed Lizzie. To remind the viewers that she isn't a psychopathic monster. That she was a confused person with a wrong understanding of the world. (They killed her because She wouldn't have been able to recuperate in that world because of the constant mental stress and traumatic experiences that she would continue to experience for the rest of her life)