The play also doesn't think we need to know terribly much about the dead Andrea. Except that he's dead. And that he and Bel-imperia were in love. And that he was killed by an un-chivalrous, cowardly act in war. And that he is very, very, impatient while waiting for the revenge he so desperately craves. Strangely enough, Andrea isn't alive at all in the play.
He's a ghost. Boo!
Even though he dies before the action of the play, we do get a ride-along with him as he tours Hell for three days. On the tour we learn that he was a lover and a fighter during his life, which leaves the administrators in Hell scratching their heads as to whether he should spend eternity with warriors or lovers. Since the hellish experts can't even make up their mind, we'll leave that alone for now.
But in the process of describing his trip through Hell, Andrea gives us an early clue about a character trait that proves vitally important:
My name was Don Andrea; my descent,
Though not ignoble, yet inferior far
To gracious fortunes of my tender youth.
For there, in prime and pride of all my years,
By duteous service and deserving love
In secret I possessed a worthy dame
Which hight sweet Bel-Imperia by name. (1.1.5-11)
Basically, he's saying he's from the wrong side of the tracks compared to Bel-Imperia. And this is a big problem for the aristocratic snobs in the play. Given his warrior credentials and his boldness for dating way over his head, we can probably assume he is courageous. And since he's not the only courageous guy of humble birth in the play (we've got Horatio and Hieronimo, remember?), maybe the point is that you're not born with nobility, you earn it.
Especially since all the nobles are total jerks.
But by far his most important trait is his impatience with the course of earthly justice. Time and time again he's antsy about delayed justice. By the end of the first act he's only seen his enemies partying down, which he responds to by saying, "Nothing but league, love, and banqueting" (1.5.4). This guy is here to see blood and so far it's been all hugging, good food, and booze.
His buddy Revenge (personified) tells him over and over to be patient, but Andrea is too urgently invested in revenge (not the personified version) to grasp the concept that vengeance is a dish best served cold. The guy just got back from three days of Hell, so you'd think he'd relax and enjoy the better weather on earth, but no.
At the end of each act we get pretty much the same routine: Andrea is anxious and Revenge is patient. The big point? Well, maybe it's that human time and supernatural time are dramatically different. Or, maybe humans shouldn't rush justice because doing so causes a big mess. We'll leave these juicy problems for you to sort out.
We should also consider Andrea's role as half of the chorus. The chorus is an old school stage convention that dates back to the ancient civilizations of Greece and Rome. In Greek and Roman theater the chorus was usually represented by a bunch of old dudes that were supposedly wise elders. These wise guys would ask important questions, narrate events that happened off stage, and provide a learned perspective on the action of the play. Here's some choric action from the ancient Greek play, Oedipus Rex. Yeah, they spook us out, too.
The Chorus in The Spanish Tragedy is really different, right? Is the big point that times have changed since ancient Greece? Because Andrea doesn't really inspire the awe and mysticism typical in ancient choral performances. Is this the best that Spain can do? Maybe.