The reality of the political situation was more complicated than the films could show. In fact, Denethor had a very good idea who Aragorn was, since Aragorn had served his father in disguise under the name of Thorongil, preparing Gondor as best he could against potential allies of Sauron.
'Denethor II was a proud man, tall, valiant, and more kingly than any man that had appeared in Gondor for many lives of men; and he was wise also, and far-sighted, and learned in lore. Indeed he was as like to Thorongil as to one of nearest kin, and yet was ever placed second to the stranger in the hearts of men and the esteem of his father. At the time many thought that Thorongil had departed before his rival became his master, though indeed Thorongil had never himself vied with Denethor, nor held himself higher than the servant of his father. And in one matter only were their counsels to the Steward at variance: Thorongil often warned Ecthelion not to put trust in Saruman the White in Isengard, but to welcome rather Gandalf the Grey. But there was little love between Denethor and Gandalf; and after the days of Ecthelion there was less welcome for the Grey Pilgrim in Minas Tirith. Therefore later, when all was made clear, many believed that Denethor, who was subtle in mind and looked further and deeper than other men of his day, had discovered who this stranger Thorongil in truth was, and suspected that he and Mithrandir designed to supplant him. ~ Appendix A (iv), ROTK
To understand why there were Stewards, and why Aragorn didn’t have a strong claim to Gondor's throne, we have to go back to the Second Age and the time of Elendil and his son Isildur, and his OTHER son Anárion, omitted in the movies. They were leaders of a band of refugees from a lost kingdom, Númenor, which sank into the sea much like Atlantis. Elendil and his followers founded TWO kingdoms in Middle-earth. Arnor was in the northwest, in the area between Rivendell and the Grey Havens (Elendil was friends with the Elves), which is where Aragorn and his people were still living thousands of years later long after Arnor had fallen. The other kingdom, Gondor, was founded south and east of the Misty Mountains, and survived even after it lost its king. Arnor was Elendil's kingdom, and Gondor was ruled jointly by his sons before the Last Alliance. When Elendil and Anárion were killed by Sauron, Isildur became king of Arnor and relinquished Arnor to his brother's son Meneldil. After over a thousand years of peace, both kingdoms started running into trouble with various enemy forces, and came under attacks from their neighbors.
Arnor came into conflict with the witch-king of Angmar, a rebellious kingdom up north ruled by the Lord of the Nazgûl. Arnor's last king, Arvedui, tried to reunite his kingdom with Gondor when the king of Gondor was killed in separate wars against the Easterlings. But the Council of Gondor ruled that Arvedui didn't have a legitimate claim:
On the death of Ondoher [King of Gondor] and his sons, Arvedui of the North-kingdom claimed the crown of Gondor, as the direct descendant of Isildur, and as the husband of Fíriel, only surviving child of Ondoher. The claim was rejected. In this Pelendur, the Steward of King Ondoher, played the chief part.
'The Council of Gondor answered: "The crown and royalty of Gondor belongs solely to the heirs of Meneldil, son of Anárion, to whom Isildur relinquished this realm. In Gondor this heritage is reckoned through the sons only; and we have not heard that the law is otherwise in Arnor."
'To this Arvedui replied: "Elendil had two sons, of whom Isildur was the elder and the heir of his father. We have heard that the name of Elendil stands to this day at the head of the line of the Kings of Gondor, since he was accounted the high king of all the lands of the Dúnedain. While Elendil still lived, the conjoint rule in the South was committed to his sons; but when Elendil fell, Isildur departed to take up the high kingship of his father, and committed the rule in the South in like manner to the son of his brother. He did not relinquish his royalty in Gondor, nor intend that the realm of Elendil should be divided for ever.
'"Moreover, in Númenor of old the sceptre descended to the eldest child of the king, whether man or woman. It is true that the law has not been observed in the lands of exile ever troubled by war; but such was the law of our people, to which we now refer, seeing that the sons of Ondoher died childless."
To this Gondor made no answer. The crown was claimed by Eárnil, the victorious captain; and it was granted to him with the approval of all the Dúnedain in Gondor, since he was of the royal house. He was the son of Siriondil, son of Calimmacil, son of Arciryas brother of Narmacil II. Arvedui did not press his claim; for he had neither the power nor the will to oppose the choice of the Dúnedain of Gondor; yet the claim was never forgotten by his descendants even when their kingship had passed away. ~ Appendix A (iv), ROTK
As you can see, Aragorn's family tried once long before to claim the throne of Gondor, but Gondor's nobles decided that their claim wasn't legitimate, and had chosen a more distant heir of Elendil instead to rule it. This was why Aragorn's family was still ruling Arnor. Once Arnor collapsed in Third Age 2106, and they weren't kings of ANYTHING, the heirs of Isildur had even less of a chance to convince Gondor of their right to the throne.
So it was that when Eárnil's son went to Mordor and was lost in Third Age 2080, leaving his Steward in charge "until his return", Gondor's rule came under the Stewardship and remained so for about a thousand years until the War of the Ring. During that time, Arnor became nothing but an empty wilderness with only a few civilized areas: the Shire and Bree. Arnor's surviving people were the Rangers or Dúnedain of the North, often fostered at Rivendell due to the old friendship between Elves and the Edain, and their distant blood-ties to Elrond, whose brother Elros had chosen mortality and become the first King of Númenor. Aragorn was Elrond's foster-son. Aragorn's challenge was twofold: help Middle-earth defeat Sauron, and prove to Gondor that, even though he was not king, he deserved the throne they had refused to give his ancestor Arvedui. His third problem was that Denethor believed Gandalf was trying to control Middle-earth by installing puppets, and resented Gandalf teaching Faramir. It did not help that Pippin brought Denethor news of Boromir's death at the same time that he revealed Théoden was now taking orders (so it seemed to Denethor) from Gandalf. Therefore, in the dramatic confrontation at Denethor's pyre — which was very different in the books, since up until that point Denethor HAD been running the defenses of Minas Tirith and only gave up after he saw the Black Fleet coming and thought defeat was certain — Denethor accused Gandalf of trying to install Aragorn as a puppet:
'But I say to thee, Gandalf Mithrandir, I will not be thy tool! I am Steward of the House of Anárion. I will not step down to be the dotard chamberlain of an upstart. Even were his claim proved to me, still he comes but of the line of Isildur. I will not bow to such a one, last of a ragged house long bereft of lordship and dignity.' ~ The Pyre of Denethor, ROTK
Faramir, he knew, would yield to Aragorn, and that irked Denethor as much as anything else. But he was not simply power-hungry. He truly believed a) that Gandalf was really trying to be in charge b) that Aragorn's claim was unlawful and c) that Gandalf, through his stupidity in entrusting the Ring to Frodo, and Faramir, through his kindness, had let Sauron recover the One Ring, thus dooming Minas Tirith and Middle-earth.