Historians are still very divided about whether Arthur ever existed or not.
The figure of Arthur is first referenced in the ninth century chronicle, Historia Brittonum. He is referred to as a military commander who fought in 12 great battles against the Saxons for the kings of Britain, but was not a king himself. He is said to have killed 960 men at a Battle of Badon.
Arthur also appears in the Annales Cambriae. It records that he was the military leader at the Battle of Badon in c.516, where he carried the Cross, and that he died along with a Medruat at the Battle of Camlann in c.537.
It would appear then that the figure of Arthur has some basis in historical reality, as a leader of the British but certainly not a king. However the authenticity of these chronicles regarding Arthur has been questioned. In particular, the entry in the Annales Cambriae most likely appeared in a later copy of the chronicle, perhaps dating from as late as the eleventh century, and therefore used the Historia Brittonum and the Welsh poem Y Gododdin (which mentions Arthur briefly) as sources. The existence of Arthur therefore cannot be corroborated by multiple contemporary sources, which does draw into question whether he was a real person. Furthermore, he is noticeable by omission from other early medieval historical works. For example, Gildas' On the Ruin and Conquest of Britain, which was written in the sixth century and so closest to Arthur's supposed lifespan, mentions the Battle of Badon but not Arthur. This is the most compelling evidence to suggest that he was no more than a figure of legend.
The King Arthur that we are familiar with is therefore likely a fabrication, fleshed out in the Arthurian genre of medieval literature which began with Geoffrey of Monmouth's History of the Kings of Britain, completed c.1136. Geoffrey claimed to have found a "very ancient book in the British" tongue and used this as his source, but even his contemporaries were skeptical. Geoffrey would certainly have consulted the chronicles that the mysterious Arthur had appeared in, but these entries are scant and required elaboration. It has been argued that Geoffrey based Arthur's character on Alexander the Great, so in that sense Arthur could be considered to have been loosely based around Alexander.