What are some similarities and differences between King Arthur and Achilles?
This is an interesting question, in that I've not heard it considered before, and, on the surface, the two heroes have little in common. Let's examine if the two legendary characters (for we have no direct evidence that either of them ever lived) do share any of the same characteristics, and what were the major differences between them. For example, there is much mystery and magic surrounding Arthur's parentage, while Achilles parentage is entirely known and acknowledged by everyone.
King Arthur becomes the ruler of a kingdom; while Achilles, though the son of a minor king of Greece (Peleus of the Myrmidons), doesn't live long enough to control a kingdom (much less become an overlord, such as Agamemnon). So Achilles, while a war leader and a prince, so to speak, is not a king like Arthur is.
Achilles is, by all accounts, by far the best warrior on either the Greek or Trojan side in this war. Arthur, while valiant in battle, certainly, is definitely secondary in martial excellence to Lancelot. Their importance to their communities is different: Achilles is the best man in battle, but not a leader of the whole army, while Arthur leads not only the Knights of the Round Table, but also the whole of his kingdom.
Arthur (at least in Geoffrey of Monmouth and in Mallory) is a resolutely Christian king (although the historical man he may have been based on may not have been a Christian at all!), imbued with the sacred task of finding the Grail and restoring order to his kingdom and goodness to the world. Achilles, though the son of a goddess (Thetis) and made supernaturally invulnerable by her (by being dipped in the River Styx) is definitely not representative of any cosmic or religious crusade. He is simply the best fighter among the Greeks, and his displeasure causes internal drama for his people. Arthur, at least as most of the stories have evolved, while certainly not perfect, was much less selfish than Achilles is portrayed.
I see little in common between the two characters, but the subject is quite fascinating. You could draw a parallel that both of the characters, in their cycle of stories (the Iliadand the various Arthurian legends) are the single most important actor. They are really the "main" characters in stories involving many people over a long period of time. They are both flawed individuals (Achilles, perhaps, more than Arthur, with his stubborn petulance which cost other people's lives) who are, nevertheless, heroes. They both have martial prowess, but one (Achilles) certainly more than the other. They both, at least for part of the story, represent virtue in some form to their people. But as far as I can determine, they are quite different characters (it would be easier to draw parallels between Agamemnon and Arthur, or between Achilles and Lancelot, for example).