Lizzie reminds Chloe so much of his childhood girlfriend Christine. Long story short - Christine and Chloe couldn't be together since he decided to become an assassin after his adopted family was killed, but Christine had a profound impact on his life.
That is partly why he was so very upset when Lizzie died. She is fast becoming very important to him, some one he feels he should look out for and protect. She is sweet, strong willed, and they way she tries to look out for Ciel reminds him of how Christine once tried to look out for him.