A native of Belgium, Achilles grows up in the dangerous streets of Rotterdam in the Netherlands. (At the start of the series, the Netherlands is under international jurisdiction during the global emergency of the Formic, or Bugger, Wars. This leads to an influx of refugee children, including Achilles.) As Achilles matures, he is noted for a brilliant mind in tactical matters and a large number of sociopathic and psychopathic tendencies, two traits that will define and characterize his short and brutal life.
Warning: Spoilers follow
Near the beginning of the series, Achilles is established as an enemy of one of the main characters, Julian "Bean" Delphiki, another urchin of Rotterdam. Both Achilles and Bean are considered for Battle School, the training ground for Earth's interstellar military; Bean is accepted, but Achilles is deemed unstable and violent, and is sent home. Bean eventually becomes an assistant to Ender and a hero of the Third Invasion.
Achilles, meanwhile, positions himself to take advantage of Earth's political circumstances after the end of the Bugger Wars. In a series of deft political and military maneuvers, he establishes a well-honed pattern. Achilles' modus operandi is to convince a set of influential leaders in a major country that his tactical genius is at their military's disposal. He uses his charisma to rise to a position of high influence in his host country's military, and prepares detailed plans for their geopolitical objectives. Once he has positioned forces in his current host country, he makes overtures to a neighboring power. At the point of his hosts' greatest confidence, he betrays them and transfers to a new country, where he begins the cycle anew. With each repetition, Achilles' host country is a larger and more powerful entity than the one he betrayed. Achilles uses this tactic repeatedly to become master of Russia, Pakistan, India, China, and eventually the Hegemony itself, the (nominal) world government.
Achilles' genius and charisma are enough to overcome even his noted history of betrayal; his sociopathy makes him unable to remain loyal to any of his masters. He is driven by a psychological need for outward strength; he must kill anyone who sees him in a position of weakness, and cannot allow anyone to attain a position of strength over him, even for a second. These character flaws cause his descent and eventual fall.
Achilles is eventually defeated and killed by Bean, with the assistance of his wife Petra, his bodyguard Suriyawong, and Peter Wiggin, the Hegemon of Earth.