Thompson's role in this movie is pretty utilitarian—he's kind of like our narrator. Instead of having much of a backstory himself, he runs around trying to figure out the meaning of Kane's final word, "rosebud." He's a man on a mission, and it's thanks to this mission that we have a movie plot.
It's only by searching for the truth that Thompson interviews the people who knew Kane and ends up piecing together the story. Thompson's questions are usually straightforward, and they tend to reflect all the same things we'd like to know as an audience, such as,
THOMPSON: When she [Susan Alexander] used to talk about Kane - did she ever happen to say anything —about Rosebud?
(Good job, Thompson.)
Of course, Susan knows nothing about Rosebud, but she goes off on a story anyway and tells us all about the tragic life of Charles Kane.
(Good job, Susan.)
Throughout this movie, Thompson has a way of anticipating our questions as audience members and saying them for us. When Kane takes Leland's horrible review of his wife Susan and finishes it, for example, Thompson asks,
THOMPSON: Well, then, how could he write that roast? The notices in the Kane papers were always very kind to her.
But Thompson is also a great reporter in the way he just lets people talk without trying to fill things in for them. In every interview he does, Thompson just says:
THOMPSON: I'd rather you just talked. Anything that comes into your mind—about yourself and Mr. Kane.
It's only at the end of this movie that we hear any kind of opinion out of Thompson on Charles Kane. The guy's been researching Kane this entire movie, and he finally makes one final statement that pretty much sums up everything we've seen so far. As Thompson says,
THOMPSON: He [Kane] had a gift for friendship such as few men have—he broke his oldest friend's heart like you'd throw away a cigarette you were through with.
More than anyone in the movie, Thompson understands the different ways that Kane was both great and horrible, admirable and despicable. So in the end, he does us a great service as an audience by putting together Kane's story and giving us a final statement on the movie.
Of course, even Thompson never figures out what we do: that Rosebud is actually the name of Kane's childhood sled.
P.S. Notice that we never really see Thompson's face in the movie? This makes him a particularly potent stand-in for the audience.