Sankarea
This is about a boy who is really into zombie stuff. His cat dies, and he goes to this abandoned building to try and bring him back to life. He finds this girl, and they become friends. She helps him find a cure, and they try it. It doesn't work. The girl lives in a rich family, but she hates it. Her dad is not good..you can watch to see what I mean. She ends up drinking some of the cure to try to kill herself. It doesn't work. Something happens later and she dies. Luckily, the cure really did work. So she came back to life. As a zombie. Sorry if I spoiled a bit too much, but this is a good anime you should watch.