Naruto didn't really despise Kakashi; no, he just wished Kakashi-Sensei treated Team 7 more equally.
. . . .
For the Chunnin Exams, when Kakashi disregarded training Naruto for Sasuke, Naruto's smile wavered just a little bit and he put up a childish front to hide his hurting. The more Naruto smiled, the less his heart was hurting because of pain and more because of the facade. Kakashi had no problem handing Naruto off to another sensei, ignoring Sakura (even though she had been disqualified from the Exams, that didn't mean Kakashi had to ignore her completely), and no problem training Sasuke for a month and that stung. After all, Kakashi-Sensei was the sensei of all of Team 7, not just the sensei of the one Sasuke Uchiha.
When Kakashi belittled Naruto's fight-of-the-year against the then bloodthirsty Gaara, Naruto was more than hurt. After all, Naruto had beaten Gaara, not Sasuke. And Naruto had enough manpower to summon Gamabunta, the Boss Toad who worked with the Fourth Hokage! Once again, Naruto smiled as wide as he could so the tears didn't slide down his face.
When Kakashi objected to Naruto learning Rasengan (his father's jutsu!), but had no quarrels teaching Sasuke Chidori (an assassin technique), Naruto was annoyed.
However, when Kakashi-Sensei remained silent when Naruto was brought back with a fucking hole in his chest, the size of Sasuke's fist, in the shape of a chidori, all the chummy feelings Naruto had for the man dissipated. Even Sakura! The girl whom Naruto loved wanted- no, begged Naruto to return Sasuke to the village, all Naruto could think was, "it's always about Sasuke. It's always has been him and always will be."
The thing that got to Naruto was the fact that his father, The Fourth Hokage (Naruto was ecstatic when he found out) was the sensei of Kakashi-Sensei. He'd taken in Kakashi when Kakashi's father had committed suicide and took care of him. Which means it was Kakashi's duty to aid Naruto when his father passed, something which he didn't do.
Kakashi was a good sensei when he actually taught, but he didn't do much teaching. He reiterated teamwork, friendships, and comradeship, but he failed to do so as he taught. By teaching only Sasuke, he ostracized Naruto and Sakura. Oh Naruto didn't hate Kakashi, he knew Kakashi-Sensei was a good man who saw the horrors of war as a mere child, Naruto just wished Kakashi-Sensei thought more of him and actually taught him.
Naruto didn't really despise Kakashi; no, he just wished Kakashi treated Team 7 more equally. After all, what was the point of a sensei when he couldn't even treat his students the same?