where did he get his name from?
The most widely accepted etymology derives it from the Roman "nomen gentile" Artorius
King Arthur
The most widely accepted etymology derives it from the Roman "nomen gentile" (family name) Artorius. 
who wrote the first book about king arthur The familiar literary persona of Arthur began with Geoffrey of Monmouth's pseudo-historical "Historia Regum Britanniae" who was this king? He was a legendary British leader who, according to medieval histories and romances, led the defence of Britain against Saxon invaders in the late 5th and early 6th centuries.
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King Arthur [SEP] Name
The origin of the Welsh name "Arthur" remains a matter of debate. The most widely accepted etymology derives it from the Roman "nomen gentile" (family name) Artorius. Artorius itself is of obscure and contested etymology, but possibly of Messapian or Etruscan origin. Linguist Stephan Zimmer suggests Artorius possibly had a Celtic origin, being a Latinization of a hypothetical name "*Artorījos", in turn derived from an older patronym" *Arto-rīg-ios", meaning "son of the bear/warrior-king". This patronym is unattested, but the root, "*arto-rīg", "bear/warrior-king", is the source of the Old Irish personal name "Artrí". Some scholars have suggested it is relevant to this debate that the legendary King Arthur's name only appears as "Arthur" or "Arturus" in early Latin Arthurian texts, never as "Artōrius" (though Classical Latin Artōrius became Arturius in some Vulgar Latin dialects).