Richard III (the king himself), was a Catholic king -- back when the real king Richard (the one Shakespeare based off of) had ruled, the Reformation wouldn't exist for another few centuries. However, by the time Shakespeare had written the play, the Reformation was in full swing in Europe, where religious groups would break off from the Catholic Church under a general umbrella term called 'Protestantism', due to differences in doctrine or disagreements with the current system. In England, it would not affect them all that much, until Henry VIII decided to form his own Protestant church (Anglican church/Church of England) to divorce his wife (a process not allowed in the Catholic Church for the reasons that Henry VIII had cited), rejecting the authority of the Pope and the Catholic Church.
The monarchy would also be a part of this Protestantism vs Catholicism debate going around in Shakespeare's time; after Henry VIII there was Mary I, a Catholic queen who began to recreate a Catholic England (and also persecution towards Protestants), then Elizabeth I, a Protestant queen who began to recreate a Protestant England (and also persecution towards Catholic). This back and forth (especially as Anglican doctrine and theology would be debated for many more years to come) would present itself in Shakespeare's work. Shakespeare himself is assumed to be a Protestant but his work also reveals many Catholic sympathies and it is believed by some that he was a secret Catholic.
That is some context from knowledge of my own; more information can be found by searching up 'English Reformation' and how it related to Shakespeare.