We get another weak leader in the form of the Spanish king. And just like with the viceroy, we learn about his weakness right off the bat. It's clear that the king is either gullible, out of touch, or corrupt when he gives equal credit to Lorenzo and Horatio for capturing Balthazar. His trusted general had just reported that Horatio did all the capturing so he should have been at least skeptical about Lorenzo's claim.
And later in the play, we find out that the king is entirely unaware that Horatio was even murdered. It seems safe to assume that Lorenzo is scheming to keep the king out of the loop, but a good king could never be manipulated so easily.
Other than that, the king's character is straight up unremarkable. The Spanish Tragedy is heavily invested in looking at how characters are impacted by corrupt or weak leaders, but we don't get detailed portraits of these rulers. Instead, we get generic rulers with a few hints to let us know they are inadequate.