In the movie, you get the sense that Aragorn doesn't want to be king... which gives more credence to Boromir's "Gondor has no king, Gondor needs no king."
Boromir is right. Why would Gondor need a king that doesn't care anything about Gondor?
Yeah, take that all you Boromir-bashers!
But what you don't find out unless you read the Appendices is that Aragorn has been defending Gondor all along under an assumed name. Of course that's for book-Aragorn only. I don't get much sense of Aragorn's past in the movie (except for the Arwen thing).
For one thing, they have Boromir immediately recognize Aragorn's name, as if everybody in Gondor knew that Aragorn was Isildur's heir, they just didn't know where he was.
PJ slips a little on portraying some of the depth of history here, I think. It's been like well over a thousand years since there's been any contact between the Rangers and Gondor and since the last King disappeared in Gondor. So when Aragorn pops up as a candidate for the throne, it would be like a claimant for the present-day British throne who's descended from the pre-Norman Anglo-Saxon Kings. Aragorn does an amazing job of convincing all these fractious aristocrats to whole-heartedly accept him as their king.