Due to the small amount of surviving Norse myths, not much is known of the origins of Loki. During his time in Asgard, however, he earned quite a reputation. Loki was not always as antagonistic as he is these days. Once upon a time he was actually more of a friendly character, certainly a troublemaker, but not a malicious one.
He delighted in playing tricks on the Gods and their rivals, the Giants equally, which would usually result in disaster. But with his quick wits and handy shape-shifting ability, he usually set things right, and often obtained items from which the other Gods benefited. In one such case, Loki transformed himself into a mare to lure away a stallion, and birthed Sleipner, Odin's eight-legged horse. In another, he tricked Dwarves to make many magical items that he later granted to the Gods. Like Thor's hammer, Mjolnir, and Freyr's ship, Skibladnir, which could hold all the Gods, but could be folded up and put in his pocket!
However, as payment for these items, the Dwarves wanted Loki's head. However, Loki tricked them, saying they could have no part of his neck. So, the dwarves sewed his mouth shut instead. It is thought that this is when Loki began to resent the Gods. He had just procured many items for them, but they did not defend him against the dwarves.
Apparently, this resentment festered, for Loki was never quite the same. He became malicious and desired to create unrest wherever the Gods went. This culminated in guiding blind Hod's hand in killing Baldur with a sprig of mistletoe. The Gods, furious, hunted Loki down and bound him to a slab of stone with his own son's entrails. A snake was set to drip poison onto his face until Ragnarok, and only the intervention of his wife kept him from constant pain.
At Ragnarok, Loki broke his bonds, and marshaled his three monstrous children, Fenrir, a fearsome wolf, Jormungand, a serpent that encircled the globe, and Hel, the mistress of the underworld, to war against the Gods. He eventually died at the hands of Heimdall. However, it appears that part of the myth was greatly exaggerated, or he managed to resurrect himself, because he is alive, and wishes to do a better job of ending the world this time.
Why was Loki not allowed to play in the Game?: Loki has, in the past, fathered three monstrous children, Fenrir, the Jourmungand Serpent, and Hel. He guided blind Hod's hand in killing the bright Baldur, and successfully brought about Ragnarok, i.e., the end of the world. He's well known for his scheming, and making mischief, and he hates anything pure and peaceful. It was felt that allowing him to play would be too much of a liability.
