Naruto’s Uzumaki last name is because its his Mother’s name. I m sure you already know that Minato Namikaze the four Hokage is Naruto’s Father. The day that Naruto’s was born and became the nine tail’s Jinchuriki and when his parents and the other ninja or you could say shinobi protecting the hidden leaf village from the nine tail’s attack many people died from the nine tail’s attack. The third Hokage wanted the villager’s to know very little about the incident since it was also decreed that it was a forbidden to talk about the incident and no ones supposed to be tell Naruto that he had the nine tail’s inside him. The third Hokage did not tell Naruto anything about his parents except that his parents died protecting the village from the nine tail’s attack. The third Hokage thought it was best for Naruto and everyone no to know that the four Hokage was his Father other wise Naruto’s life would of been in Danger if anyone found out and there would also be a protest or someone sort of riot or something that would destroy the balance of peace in the hidden leaf village and no doubt that Naruto’s life would be in the Danger and might have died if something like that ever came again. This is why his name last name is the uzumaki and its also because his mother kushina is from the uzumaki clan’s from the land of eddies like the Uchiha the uzumaki clan is also and extinct clan only a few sole survivors scatted all over some are still living or hidding in a different village or nation. You learn about that the nine tail’s incident in the 1 Naruto season in the very beginning. They tell you about the decree and how its is forbidden to talk about the nine tail’s incident and how no one can tell Naruto that he had the nine tail’s inside of him. You learn about that reason in the Naruto Shippuden when you first see the four Hokage he tells us the reason well or what he believe’s is true since the third Hokage didn’t tell Naruto much of anything.You learn the rest about his mom and her clan the uzumaki clan and about the nine tails incident later on in Naruto Shippuden. The Third Hokage named the boy uzumaki in order to the hide his connection with Minato Namikaze who had a large share of enemies especially in the Iwagakure. This is made perfect sense if Naruto’s mother was some nobody. However it was later revealed that his mother Kushina was a famous and powerful kunoichi, Jinchuuriki and a member of a very famous and powerful clan to boot. There fore in this Hiruzen Sarutobi’s decision wasn’t smart at all.The changing of the name was done by the Third Hokage, Hiruzen. He felt that Naruto would be the hunted down by the enemies of Minato Namikaze, for example, Kumo and Iwa. So for Naruto’s own protection, they change his name from Namikaze to Uzumaki, because no one except for Jiraiya, Hiruzen, his wife Biwako, and Tsunade knew about the marriage of Kushina Uzumaki and Minato Namikaze. So no one knew that Naruto was a child there and because of this fact, people would not make the connection between the Kushina Uzumaki and Naruto. Naruto didn’t even have the customary and red hair of the clan, a fact that be worked in Hiruzen’s favor, as people would think the that Naruto was a clan less orphan.
Actually the Uzumaki Clans was wiped out as well. However it happened way before naruto’s was born. During the third ninja war, other villages except hidden leaf, feared the clan’s sealing jutsu’s (they could seal a monster with make just a few hand signs). So they ambushed the village’s and killed everyone. (don’t get confused here! The Uzumaki Clan’s had their own village named Uzushio-gakure no sato) How ever there have been some survivor’s such as Nagato parent’s, Karin parents and Kushina Uzumaki parents. If you want to know watch the episode, it’s the 247th.