"Viceroy" is a fancy word for a stand-in king. In large empires, overworked kings would often delegate powers to viceroys, who basically served as kings in their respective realms. All you really need to know for the purpose of our play is that the viceroy has eminent power within his sphere of influence.
But boy does he stink at being viceroy: he loses the war with Spain; follows a bunch of wrong information that leads him to sentence an innocent man to death; and whines himself into despair that distracts him from his duties. We get an early clue that the viceroy is a spineless ruler when he says,
Then rest we here awhile in our unrest,
And feed our sorrows with some inward sighs
But wherefore sit I in a regal throne?
This better fits a wretch's endless moan. (1.3.5-9)
Hello? There's no rest for viceroys. This guy clearly lets his emotions get in the way of his national duties. A Renaissance audience would immediately see the viceroy as a weak ruler. Especially when he questions why he should even sit on a throne. Sheesh.
His weak leadership skills and his tendency to wallow in his emotions invite us to see all the problems in the play as systematic. In other words, you may want to think about how weak leaders allow for an unjust environment. How might strong leadership have prevented tragic outcomes in the play? Does the buck stop with the viceroy?