Asch the Bloody is a God General of the Order of Lorelei. He can be arrogant and rebellious, and almost always seems angry. He has a tremendous amount of pride and cares deeply for his country and the people around him. He is also an extremely talented cook. Despite his smart and calculating exterior, Asch has moments of incredible goofiness. At one point he masquerades as Luke and is asked (by someone who had met Luke a few days previously) how his hair had changed from short to long in such a short time. Asch's improvised excuse was, "My hair grows fast."
Most of what Asch does is the "behind the scenes" work - though works towards the same goal as Luke and friends. However, most of the time, he outright refuses to work with them. At the end of the game, he even fought Luke over who would fight Van.
Asch openly despises Luke as the person who replaced him. He hates how Luke considers himself (as the clone) to be the inferior one, as not only is he a copy of Asch (and Asch has a tremendous amount of pride) but he was the one who essentially usurped Asch's life with no effort whatsoever. He also considers them rivals in swordsmanship as they are both pupils of Van.
Although he thinks that Luke is supposed to be his replacement, and thus the same as him, there are many things in the game that say otherwise. Perhaps the most obvious of these is that Luke's hair is light at the ends, and Asch's is darker. Their clothing is also like this, with Luke's white and gold contrasting with Asch's black and silver. Yet another difference between them is that Luke is left handed, and Asch is right handed. Also, as Luke gradually changes throughout the game, Asch always stays the same.
Luke and Asch fought (due to Asch's stubbornness) at the end of the game to settle the issue over who the "real" Luke was. In the end, Luke did win, yet Asch still thought of Luke as his replica. One of his last actions was to take his name back as his own, as a small step in accepting Luke, and acknowledging that he wasn't inferior.
Upon his death, he gave Luke some of his power, allowing Luke to use second-order hyperresonance (and to fully control it at that). It gave Luke the power he needed to finally defeat Sync and Van.