In Inglorious Basterds, just as Stiglitz was introduced, we are informed of his previous run-ins with the German army for killing 13 Gestapo officers. What I've always seemed to miss was Stiglitz's motivation for the murders. Did Stiglitz kill them because he was far too disillusioned with the Nazi ideology, or was he simply a psychopath who just happened to put his deranged tendencies towards achieving something that is socially acceptable?
As far as we know, it's never clearly said or mentioned somewhere. The best and closest I could find to explain it is this:
Hugo Stiglitz is quiet, and doesn't seem to enjoy social interaction. He is a German who hates Nazism and Nazis. He seems to get along with the Basterds as he is seen laughing with them and cheering Donny on when he's using his bat. Hugo also seems to have a personal hatred for the Nazis based on a flash back showing him being tortured by one prior to his enlistment.
Like many Germans at the time, he hates Nazis, and like many Germans, he fought them. We can suspect some kind of mental disorder though, as he acts like a psychopath.