Sue is a verb, I don't think "a sue" is used. It means you start legal proceedings against someone, e.g. you intend to go to court.
* Samsung and Apple are always suing each other
Claim can be a verb or a noun:
verb: to say something is true (without proof) OR to say you are entitled to something
noun: a statement that something is true OR a demand for something you think you are entitled to
* He claimed that he had been sober
* The court had denied their claims to asylum
So both "sue" and the second meaning of "claim" are related to court stuff, which perhaps causes the confusion.