Mainly in that sappy scene with Arwen before (in front of) the shards of Narsil where he laments his weakness as being the same as Isildur's. He is definitely questioning his kingly (I almost typed kinkly) qualities at that point. But I'm with you on Aragorn overall. He is growing as the story progresses and that is fine....
I think it is only about the ring... and foreshadows the end scene where he passes the test with Frodo. I don't see him questioning that he will be king (which in the movie we don't know yet), only whether he will not fall to evil on his way there, whether he will try to use the ring.
In the book I think Aragorn is whiny at Bree... I just can't stand this line in the book: "But I must admit,' he added with a queer laugh, 'that I hoped you would take to me for my own sake. A hunted man sometimes wearies of distrust and longs for friendship. But there, I believe my looks are against me.'"
Aragorn needs a hug!
I don't think he's questioning whether or not he could be King, but is wondering whether the weakness of Isildur was passed to him through the bloodline along with his title.