All trolls are made of metamorphorical rock. The Diamond King is a Troll born not of jasper or beryl or bluestone or granite, but pure diamond, and as such his destiny is kingship. As diamond trolls, they are capable of regulating the temperature of their brains by reflecting all heat. The troll brain being a lump of impure silicon, troll intelligence is lowered by increases in temperature. Diamond trolls' ability to regulate temperature makes them very intelligent, and intelligent in all environments.
Real diamond is pure carbon (the element that we squashy people are based on), seemingly denying diamond trolls the ability to have a silicon-based brain. But Mr Shine's discussion with Vimes in Thud! confirms that the metamorphorical rock of trolls only "looks like certain types of rock". This gives them some of that rock's properties or abilities, but they are metaphorically - not literally - made of mica, shale, diamond, brick and so on.
The current Diamond King is Mr. Shine, who has spent most of his life living undercover in Ankh-Morpork, though he reveals himself at the end of Thud!. Another Diamond King brought the trolls to Koom Valley, and died in a cave playing Thud with B'hrian Bloodaxe after the "Battle", having recorded his voice on the Cube.
In Raising Steam, it is revealed that not all diamond trolls are destined to become king of the trolls. It appears that if there are at least two diamond trolls on the Disc at the same time, then superiority travels with seniority, and the oldest troll becomes king. This is apparent because Thunderbolt, the Disc's first and only troll lawyer, is the nephew of Diamond King of Trolls, and is diamond himself. However, he has not seized the throne. Mr. Shine told Sam Vimes that diamond trolls are extremely rare, so perhaps the current situation is new, and Thunderbolt prefers not to be the new King.
Additionally, in Raising Steam, it is stated that diamond trolls do not tell lies. It is unknown if this is an actual product of their physiology, or if this is simply a code of ethics that diamond trolls adhere to.