how would I have to format my data to make it work with Keras?
Your training labels in the output layer should be a binary vector that is 1 for class which is present and 0 for class which is not. For example, let us assume you have 3 classes of genres - comedy, romantic and horror. There are many ways to make it and Scikit-learn has a method which makes it very easy which I show below.
>>> from sklearn.preprocessing import MultiLabelBinarizer
>>> mlb = MultiLabelBinarizer()
>>> y = mlb.fit_transform([[0,2],])
array([[1, 0, 1],
[0, 1, 0]])
I initially considered using a softmax layer as my output layer, but since a movie can have multiple genre labels, how should my output be?
This is a simple Keras example I suggest.
>>> from keras.models import Sequential
>>> from keras.layers import Dense, Activation
>>> model = Sequential([
>>> model.compile(optimizer='rmsprop', loss='binary_crossentropy')
>>> model.fit(X_train, y_train)
Refer this for more info. I used sigmoid because it is better for multilabel classification.