Revenge is revenge personified. Yep, it's that simple.
And by a personification, we mean he's the idea of revenge in the flesh. Being the personification of an idea makes him pretty single-minded. There's not much room for reflection or, well, anything else. But, hey, he does talk real fancy:
Then know, Andrea, that thou art arrived
Where thou shalt see the author of thy death,
Don Balthazar, the Prince of Portingale,
Deprived of life by Bel-Imperia.
Here sit we down to see the mystery,
And serve for chorus in this tragedy.
Can't you just hear a big, deep dramatic voice saying this?
Anyway, Revenge is all about seeing bad guys killed for doing bad things. He accompanies the ghost of Andrea to the land of the living, where the two serve as chorus to the play. Where Andrea is hotly impatient to get his revenge, Revenge is patient, as characterized by the great line, "Nor dies Revenge, although he sleep awhile; For in unquiet, quietness is feigned" (3.15.23-24). So yeah, Revenge's biggest character trait is patience.
The big lesson? Well, it could be a warning to all us humans that we never really get away with anything. Because as soon as we think we're in the clear, ol' patient Revenge is there to see that we get it in the end. Or maybe he's just a really chill dude. Never mind, he's there to remind us that he's always there. So behave.
In the olden days (and we're even talking before this play), there were all kinds of personifications in plays. You'd get guys like Greed and Vice walking around getting all greedy and vicey all over the place. Personifications were used to examine human traits in their purest form. As such, they often served as warnings for traits we should avoid.
The tradition of personification goes back to the kinds of stock characters found in medieval Morality plays like Everyman. In Morality plays, the protagonist meets personifications that try and lead him down the path of destruction. By the time of The Spanish Tragedy, these characters were going out of fashion. But maybe Kyd uses the convention to suggest that Revenge leads to destruction. Or maybe he just thought Revenge was an eerily disturbing presence from the past.
Whatever the case, just like in fashion, going retro is meant to make a statement. So why is this play dressed in old clothes?