This article is from the 2006 series of SMBZ and is no longer canon.
Omega's goal was to destroy all of Eggman's other robots for revenge and prove his superiority to them. He possessed a great deal of physical strength and a wide array of weaponry in his arms and torso, mostly gatling guns and explosives. Omega had become one of Shadow's closest friends, which probably explains Shadow's bad attitude.
Ultimately, Omega was destroyed by one of Eggman's creations, Mecha Sonic. It should be noted that in the Archie Comics and in the games, Omega was shown capable of possibly defeating Shadow (Sonic Universe 3 and Sonic '06) and considering Shadow's common equivalence to Mecha Sonic, it is a confusion to how Omega was possibly destroyed by Mecha Sonic. It should be noted, however, that Shadow has been shown to be much faster than omega, and Omega's portrayal in the comics are not canon to the game series, and Sonic '06 displayed circumstances that made Shadow's defeat inconclusive, so they may not be valid in concluding Omega's strength in the Series. Sonic '06 was also continuity resetted due to a time paradox that erased the game's events.