Will eye cancer spread to other parts of my body?

Depends. It depends on the type of eye cancer you are talking about. For example, choroidal melanoma can spread for example, most commonly, to the liver. It doesn't mean that it will but it can.
Possible. Like all cancers, eye cancer can metastasize and grow in other parts of the body. That is why screening is important and why you should notify your ophthalmologist if you notice growths on the lids or changes in the vision.