Without trying to diminish newer answers, /u/kielowskifans already explained this in another thread.

Hitler's Austrian background was not that much of a problem for him as there was a sizable number of Germans in the 1920s and 30s that considered German-speaking Austrians to be ethnically German.

I see that this has been answered with a link to a somewhat related post, but I'd like to try and give you a more concise answer to what you're looking for.

While Hitler was an Austrian-born man, there are a certain set of circumstances that surround the scope of this question. After the assassination of the Archduke, Austria and Germany knew that it was a given that war was coming soon. Hitler, during the four or so years leading up to the war, had spent a lot of time in Vienna. Vienna is where Hitler did the majority of his learning, growth, and development as a person and the city deeply shaped his political views. While Hitler was an Austrian, he had a fascination with the German people and the German country. Though one could make the argument that Austrians are Germans, Hitler saw the Austrians, though powerful, as slightly weaker than the German people due to the types of individuals that held government positions and positions of power in Austria (Jews, immigrants, etc.)

When WWI broke out, Hitler was called in for a physical examination to see if he would be drafted into the Austrian army. Many historians contend that he failed his examination intentionally so that he could instead volunteer for the German army. Regardless of the facts surrounding his failure at the drafting exam, Hitler did relocate to Bavaria, Germany where he volunteered for the army.

While military service in the present era has the potential to earn a person citizenship, this wasn't the case for Hitler. Hitler, throughout the war, served with distinction, was injured and almost killed on multiple occasions, and even earned the German Iron Cross First and Second class! Though he only made the rank of corporal, primarily because he turned down promotions that would take him from the action, he put his life on the line for the German people and the Kaiser, but after the war he was mostly tossed aside.

Following WWI, Hitler was made into a spy for the Weimar military. His job had him spying on political parties that were deemed radical and potentially dangerous. Essentially, he was there to decide whether or not a group posed an actual threat to the very shaky stability that existed. It was here that he found the Germany Workers Party (DAP).

After years of building the party and other circumstances, Hitler's hand was forced and the infamous Beer Hall Putsch occurred. The Putsch was a planned revolution that was supposed to sweep Germany to take the government by force.

Though the Putsch failed, something interesting occurred. Hitler was put on trial and he was convicted of treason in 1924. Hitler, however, wouldn't gain German citizenship until 8 years later in 1932. His citizenship came about a whole one year before he was elected Chancellor of Germany. This begs an interesting question as to how a person without citizenship could be convicted of treason. This factor alone was, ironically enough, one of the reasons Hitler even gained his citizenship in the first place.

My information comes from various books, but a great source for this information and the life of Hitler in general is: Hitler- A Biography by Ian Kershaw.