Tyrion Lannister is (personally) one of the best anti-heroes after Walter White (Breaking Bad). He has walked on one of the toughest paths in the whole 'GoT' universe and his path of redemption and self-righteousness was neither expected nor feasible by the audience. People perceived him to be a rich brat who was useless both in the army as well in the royal castle. He was a man who only sought after the royal family's benefits without ever putting in the efforts to help his family. He had the excuse that his family hated him and always treated him like an outsider but he wasted his youth and wisdom in prostitutes and alcohol until life forced him to act maturely.
His 'trial by combat' put him under the crosshairs of his own siblings for the murder of King Joffrey. It is quite a complicated story but when he lost everything and was on a low, life brought him an opportunity that he never expected and that opportunity helped him reach his full potential. An unjust & fair advisor plus a master tactician. When the audiences first met Tyrion, they never expected him to hold such an auspicious position and responsibility but he did and his transformation into the fine man he is was a long internal battle. He fought with himself, his demons & his vices to change himself.
I am a sucker for redemption stories and how Tyrion blossomed into a key character of the show till the last episode was one of the best aspects of the great show. Only great writing and acting could have made it possible!