Question: My antagonist is cunning, manipulative, and won't hesitate to resort to murdering innocents to achieve his way. But he has a granddaughter whom he truly does love. Will this make him seem weak? After all he is the villain. But this question doesn't just apply to my villain. If a villain has a soft spot for something does that make him a weak villain?
No, it just makes him more human.
If you want to write a cartoon villain, you might get away with making him thoroughly evil, but that would be a villain with no depth.
Human being are complex creatures who are seldom black and white. For instance, even Hitler loved his wife. The nastiest people in the world often have loved ones, friends, etc. One assumes they must have some good qualities in order to retain their supporters. (Of course, there are psychopathic loners too.)
Sometimes contrasting elements in a personality can work to your advantage. The fact that Ernst Blofeld, the villain in the James Bond films, had a pet cat he was always stroking made him seem even more evil rather than less. Consequently, this trait has been copied ever since (in characters such as Dr. Evil, Voldemort with his snake, etc.).
There are also villains whose tragic background makes us feel sympathy or empathy for them. The great thing about these characters is that they put the reader into a moral dilemma. We can see that what they are doing is wrong, but we feel sorry for them at the same time.
Of course, just because the villain has a granddaughter does not mean he has a loving personality or inspires love in her--even if he wishes he could. As the writer, you can control how much the reader empathizes with the villain by how you write this relationship.