Please provide a character sketch of the emperor of Lilliput.
The emperor of Lilliput is not only a small man in terms of physical stature, but he is also small in the figurative sense of being petty. This pettiness is on full display when he becomes angry with Gulliver because Gulliver is unwilling to help him assume total control over Lilliput's enemy and neighbor, the island of Blefuscu. Gulliver helped Lilliput to win the battle, but when he refuses to assist Lilliput to conquer the Blefuscudians completely, the emperor grows angry and forgets the good services Gulliver has already done for the kingdom.
Not only is the emperor petty, but he is also deceptive. When Gulliver goes away to visit Blefuscu, the emperor seems as though he remains Gulliver's friend; however, the emperor -- along with some of his more malicious courtiers -- plot how to rid the kingdom of their giant (and to the emperor's mind, expensive) liability. They accuse him of treason and plan to execute him upon his return to Lilliput.
The emperor of Lilliput proves himself to be savage, merciless, and selfish. He clearly thinks highly of himself and much less of practically everyone else. He only really sees Gulliver in terms of what Gulliver can do to bring him more power, and he is unscrupulous when it comes to anyone who defies or challenges him.
By sketch, I am assuming you mean a literary sketch of the Lilliputian emperor. As you know, the Lilliputians are physically tiny people, and Swift plays upon their physical size by linking that to their "smallness" in character.
The emperor of Lilliput is egocentric and acts much like a despotic ruler. He demands that his officials walk on tightropes and perform other dangerous acts to obtain and keep their jobs. They oblige, even though these feats have nothing to do with their jobs, just to keep the emperor happy.
The leader also shows a lack of judgment in allowing a fight between two nations to rage on about the proper method to crack eggs. This long, insignificant battle has raged for years and cost many lives.
In order to be freed from his bonds, Gulliver must agree to a long list of items, including defending Lilliput at all costs. However, the emporer shows his lack of appreciation for Gulliver both saving his wife's life in the burning castle and for repelling an attack on Lilliput by impeaching him.
Gulliver eventually is freed from his stay with the small-minded Lilliputians and goes on to bigger and better places.