Gilgamesh the King by Robert Silverberg

From the publisher (source: The Internet Book List):
Two-thirds god and one-third human, Gilgamesh is a giant among men and a formidable warrior, even as a boy. When his father the king of Uruk dies, Gilgamesh is forced into exile by the newly crowned Dumuzi, jealous of his prowess and fearful of his intentions.

In neighboring Kish his fighting skills are honed to perfection, and when in time Dumuzi dies, Gilgamesh returns to be proclaimed king by the wily priestess Inanna, goddess of beauty. Together they rule Uruk, and prosperity descends upon the land.

However, the kingship is not enough to satisfy Gilgamesh's gargantuan appetite for adventure, and his boredom is only relieved by the coming of Enkidu, a strange wild man who proves the king's equal in combat. The two become closer than brothers, but when Gilgamesh incurs the wrath of Inanna, the gods conspire to tear them apart, and for Gilgamesh all that remains is his search for immortality.

This book is part of the "Gilgamesh" series.
Here are some other books from this series: