Lurtz is one of the first Urak-hai to ever be created by Saruman. From the start he shows his brutality when he kills one of the birthing Orcs by strangling it with his bare hands. This show of strength, with a higher level of intelligence than most, brings Lurtz to the position of a "right hand man" for Saruman. Saruman then fills his head with hate of all living people and makes sure he is completely bent on doing any command he issues.
Lurtz is then appointed to lead the Urak-hai warriors in their attempt to find the Hobbits as they know one of them has the One Ring. He helps them get ready as they're suited up and marked with war paint, white in the shape of a hand to represent their allegiance to Saruman.
During their hunt for the Hobbits he also shows excellent scouting skills, pointing the way for the other Urak-hai. It becomes a chase as the remaining Fellowship attempt to escape by the river. They eventually meet up at Amon Hen, first running into Aragorn.
Lurtz goes on to find two of the Hobbits, being Perry and Merry. They're guarded, however, by Boromir who has recovered from the evil pull of the ring and leaps to their defense. He slays a bunch of Urak-hai before Lurtz, standing at the top of the hill, shoots an arrow into his chest. Boromir bravely fights on, stubborn in his attempt to prove himself before he dies. Lurtz shoots him two more times before he finally gives up.
The Urak-hai run off with the Hobbits slung over their shoulders as Lurtz walks up to Boromir and points a cocked arrow at his head. Aragorn arrives just in time, tackling Lurtz and sending the shot off wide. They then join swords in battle, with Lurtz at one stage using his shield as a weapon. Aragorn manages to stab him in the leg with a ceremonial knife but he simply pulls it out and licks it to show how tough he is. Aragorn eventually triumphs, cutting off one of his arms and then lunging the sword into his chest. Refusing to die, Lurtz then pulls himself along the sword, growling. Aragon then wrenches it back out and cleaves his head off, effectively ending Lurtz's short-term life.