Gaal is the narrating character of "The Psychohistorians," but he's not the protagonist. Instead, he's the chronicler character, like Watson is for Sherlock Holmes. So we witness the entire the story from his point of view, even though he doesn't actually do much but watch the events unfold in front of him. The result? There are two distinct parts to Gaal's character.
The Math Genius
First, he's a math genius.
The core of "The Psychohistorians" deals with psychohistory, a super complex mathematical system that can predict the future. In Chapter 5, Seldon demonstrates psychohistory for Gaal with a long string of equations that we never see. (Lucky for us.) Here's what it looks like when Seldon does the math:
[Seldon] proceeded. As each item was mentioned, new symbols sprang to life at his touch and melted into the basic function which expanded and changed. (I.4.23)
Why leave it at that? Because most of us aren't mathematicians, so even if Seldon showed the work like a good math student, we wouldn't know what to do with the numbers.(Probably wonder what all those letters were doing there.)
That's where Gaal comes in. He becomes our surrogate math mind. Gaal is a reliable narrator in this regard because of his background in mathematics (he owns them fancy advanced degrees). He needs only to say he checked the numbers and approved them, and that's good enough for us. We don't need to see the math equations. Instead, we can just trust Gaal's knowledge and move on with the more important stuff, like—oh, let's think—the story.
Also, if he wasn't a super-smart math nerd, then why would Seldon even invite him to work for him?
The Country Bumpkin
But Gaal also needs to be an idiot at the same time. Okay, not an idiot—that's pretty harsh—but certainly not wise to the ways of the world.
He needs to be what we call "a fish out of water" character. This type of character is one who finds himself out of his element. Think a country boy suddenly thrust into the big city, upper-crust socialite scene (Pip from Great Expectations, anyone?). The advantage to such a character is that they need everything explained to them, meaning everything is explained simultaneously to the reader.
The same goes for Gaal. Gaal has never left his home planet. He has never been to Trantor, and he doesn't understand how the Empire really works since his head has been buried in math books for most of his life. As a result, he doesn't know anything about the universe at large—who really controls it, how it really works, the economic policies, the political aliments, nothing. The man can't even hail a taxi.
That's convenient for us as readers. Through Gaal's eyes, we discover Trantor and learn how it works. When Seldon has to explain the Empire's woes to Gaal, he also explains it to us. The same goes for when Avakim tells Gaal about how Empire law really works. We discover and build an understanding of the universe with Gaal because we're just like him. If you think about it, in universal terms, we're all country bumpkins too.
So, that's the character of Gaal. Narrator, math genius, and country bumpkin all in one: the perfect guide to the universe of Foundation.