I believe that Philip actually did claim the English throne, on the basis of his descent from John of Gaunt. My understanding is that he was planning to give England to his daughter Isabella, the one whom he later promoted as a potential Queen of France and who later became co-ruler of the Spanish Netherlands with her husband.
I had thought he basically just wanted Isabella on the throne as queen in her own right. Were there any plausible English Catholic candidates? The most plausible candidate appears to be Lord Strange (Ferdinando Stanley, later 5th Earl of Derby), who was descended from Henry VIII's sister Mary's younger daughter, Eleanor Brandon, and who apparently kept his religion ambiguous in order to attract possible support from all quarters. Unfortunately, he's already married. Other Tudor-descended candidates - James VI of Scotland and Lord Beauchamp - and the Yorkist possibility - the Earl of Huntingdon - were all Protestants. Arabella Stuart obviously could not marry Philip's daughter. There's also Lord Stafford, who descended from the Beauforts, but he seems pretty obscure. I'm not sure any of these people would make much sense, which is why Philip was claiming the throne for himself.