CNN Multi-class vs Binary Class Image Classification
I would train one classifier with 3 classes: 

cat, dog, neither


Use cross entropy as your cost function.

There are many examples of this online. If you Google MNIST neural network you'll find many simple examples. Or you can take one of Google's trained networks and retrain the last layer.

Here is a convolutional being trained on the Cifar set of images.

Using a pre-trained network to tell two house cats apart.