[Leaf]

Originally Posted by Kuruharan

I am currently thumbing through the first couple chapters where Aragorn appears of Fellowship just to see if there are any noteworthy moments of "claiming" that happen in these chapters. I don't think that there will be much useful for this topic in these chapters.

I did the same and here are some related passages I could dig up.

There's the poem in Gandalf's letter to Frodo in the chapter Strider:

Originally Posted by Strider

[...]Renewed shall be blade that was broken,
The crownless again shall be king.

Aragorn's explanation

Originally Posted by Strider

'But I am Aragorn, and those verses go with that name.' He drew out his sword, and they saw that the blade was indeed broken a foot below the hilt. 'Not much use is it, Sam?' said Strider. 'But the time is near when it shall be forged anew.'

We get an idea of Aragorn's royal descendancy and his intentions from the very beginning. Then there's a litte bit in the Flight to the Ford:

Originally Posted by Flight to the Ford

'Where did you learn such tales, if all the land is empty and forgetful?' asked Peregrin. 'The birds and beasts do not tell tales of that sort.' 'The heirs of Elendil do not forget all things past,' said Strider; 'and many more things than I can tell are remembered in Rivendell.'

Finally Aragorn's role and background gets explained in detail in The Council of Elrond:

Originally Posted by The Council of Elrond

'And here in the house of Elrond more shall be made clear to you,' said Aragorn, standing up. He cast his sword upon the table that stood before Elrond, and the blade was in two pieces. 'Here is the Sword that was Broken!' he said. 'And who are you, and what have you to do with Minas Tirith?' asked Boromir, looking in wonder at the lean face of the Ranger and his weather-stained cloak. 'He is Aragorn son of Arathorn,' said Elrond; 'and he is descended through many fathers from Isildur Elendil’s son of Minas Ithil. He is the Chief of the Dúnedain in the North, and few are now left of that folk.'

Aragorn's response to Boromir

Originally Posted by The Council of Elrond

'[...]For the Sword that was Broken is the Sword of Elendil that broke beneath him when he fell. It has been treasured by his heirs when all other heirlooms were lost; for it was spoken of old among us that it should be made again when the Ring, Isildur’s Bane, was found. Now you have seen the sword that you have sought, what would you ask? Do you wish for the House of Elendil to return to the Land of Gondor?'

Originally Posted by The Council of Elrond

Aragorn smiled at him; then he turned to Boromir again 'For my part I forgive your doubt,' he said. 'Little do I resemble the figures of Elendil and Isildur as they stand carven in their majesty in the halls of Denethor. I am but the heir of Isildur, not Isildur himself.[...]'

Originally Posted by The Council of Elrond

[...]'But now the world is changing once again. A new hour comes. Isildur’s Bane is found. Battle is at hand. The Sword shall be reforged. I will come to Minas Tirith.'

In conclusion it seems to me that Tolkien was firstly trying to establish the legitimacy of Aragorn's claims. After that Aragorn gets to be more assertive about them.