He descends through both of the sons of Elendil - in direct descent through Valandil, youngest (and only surviving) son of Isildur, and somewhat more distantly from Anarion through Firiel, daughter of King Ondoher.
But is this enough? The last King of Gondor died in T.A. 2050 - it would be almost a thousand years before Elessar took the throne, in T.A. 3019, and three thousand years since any heir of Isildur had ruled Gondor.
When Onodher and his sons died, Arvedui, another heir of Isildur and the husband of Firiel, tried to claim the throne and was denied, in favour of Earnil II, a descendant through the male like of Anarion and cousin of the fallen King. So at the time, no, it was not considered enough.
Although Aragorn's right to the Kingdom of Arnor, the Kings of which he descends through, is recognised by the fact that he is Chieftain of the Dunedain, who are more or less the remnants of the Northern kingdom, should his distant connection to a royal bloodline be enough to give him a crown which has gone unused (and mostly unmissed) for centuries?
Aragorn's experience is almost entirely military. Although he has, I'm sure, a fine education from Elrond, he spent most of his time before the War of the Ring wandering the wilderness. When he did come to Rohan and Gondor under the guise of Thorongil, again, he seemed to serve in a mostly military capacity, as in Gondor when he led armies against Umbar.
Does this prepare him to oversee the day-to-day workings of Gondor, of Minas-Tirith and its far-flung lands? Of course not. Of course, there are many people (such as Faramir, for example) in Minas Tirith who have experience in these matters, in some cases (such as Faramir, for example), who belong to families that I'd argue were no less noble than the house of Elendil.
But why should they settle for second-best? Let Aragorn Elessar be King of Arnor, and Faramir be Steward of Gondor; Aragorn's home has always been in the north, not within the confines of Minas Tirith. Elessar is suited for times of war, and can aid them in such; but the House of Hurin, which has ruled Gondor unaided for a thousand years, have guided their people through both war and peace.
Oh, just have at.