Aragorn was not exiled... he was from the northern kingdom or Arnour (I think that is how it is spelled). That kingdom was conquered roughly around the time that the last king of Gondor died and the stewards took up the scepter. The reason he was wandering is because he is a ranger (the dunedain of the north). As heir to the northern kingdom, he was their cheiftan. In order for him to be king, he would have to be accepted by Gondor (there are issues with choosing an heir in the past... read the appendix, it is in there). So, he is the rightful heir cause he is a descendant of Isildur, but he is not a descendant of the last king of Gondor (he had no descendants, otherwise they would have been king). He actually serves in Gondor earlier in his life as a captain to the steward Ecthelion. He went by the name Thorongil at that time.
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