Upon discovering the brutal murder of his son Horatio, Hieronimo is understandably grief-stricken. Not only that, he's full of hatred towards his son's killers and vows that he will seek revenge upon them. But his wife Isabella tells him that revenge is wrong, and that the punishment of Horatio's murderers should be left up to God and the law.
At first, then, Hieronimo tries to seek justice through the law. As the Marshall of Spain, he often acts as a judge in the law courts, and so going by the book is the appropriate means of seeking redress for someone in his position. Those who administer the law should not be violating the law by committing acts of vengeance upon their enemies, no matter what they may have done to them.
Hieronimo also has to bear in mind that seeking revenge upon his son's killers could have serious political consequences. The two men, Lorenzo and Balthasar, are both senior members of their respective royal families. For Hieronimo to kill these men would, therefore, have severe political repercussions that could cause irreparable damage to relations between Spain and Portugal.
So Hieronimo tries to seek justice through the law. But he soon realizes, much to his anger and frustration, that this will avail him nothing. In his quest for justice, Hieronimo is blocked at every turn, largely for political reasons. At the same time, Hieronimo, in his capacity as a judge, is still expected to dispense justice to others and uphold the integrity of a legal system by which he's becoming increasingly disillusioned.
Driven out of his mind by a lingering sense of injustice, it's just a matter of time before Hieronimo does what he originally intended to do: seeking revenge.