Sometime between World War I and the year in which Hellsing takes place, Schrodinger was created. It is known that he was created by the Millennium scientist known only as "Dok". Other than that, incredibly little is known for sure about the capricious cat-boy. His true age, the extent of his powers, and what he was before are all unknown quantities at this time.
Schrodinger tends to act like the small, malicious, irreverent child he looks like. He wears a Hitler Youth uniform, often talks back to or disrespects his superiors (though he can threatened into line), brags about his prowess, lounges around in an undisciplined fashion, forgets his instructions, and complains about what small duties he's assigned. This is at least partially a facade. Schrodinger is in fact more canny than he lets on, more aware of the true situation than he pretends, and probably considerably older than he appears. If he has even the slightest amount of compassion, caring, or loyalty to his allies, he hides it well. While not as openly psychotic as many members of Millennium, he takes childish glee in acts of torment and slaughter, including those inflicted on his so-called allies.
Schrodinger is apparently not a vampire; precisely what he is is unknown, though it is feline in nature. His most notable displayed power is also summed up in his catchphrase: "I am everywhere und nowhere." Schrodinger, like the cat he is named after, appears to both be in a place and not be in a place at the same time. He can also go from one place to another instantaneously, and these places are not necessarily physical, as he has demonstrated the capability to pop up unexpectedly in the minds of others. This ability also makes him extremely hard to kill, as even blowing his head to bits does not cause him anything other than the most mild of inconvienences; his body merely vanishes as he jaunts back home, none the worse for wear. It is quite possible, even likely, that Schrodinger's abilities are even more wide-ranging than this, but he has not displayed them.