How did Hitler become Chancellor of Germany when he was Austrian and born in Austria?

He joined the German military and served during WWI. Joining another country's military has long been used as a quick way to gain citizenship in that country.

Great question.

Hitler was a stateless immigrant to Germany after he surrendered his Austrian citizenship around 1925 to avoid deportation. Hitler desired to attain political powers in Germany more than anything. In 1932 Dietrich Klagges, then Germany's Minister of the Interior, appointed Hitler to an government position with the Braunschweig government, automatically making him a German citizen.

Now, understand that Hitler always wanted to unite the Ayian race, bring all those Nordic, Germanic, if you will, folks back "home" into what he envisioned the Great German Reich, also known as the 3rd Reich, following the Greek and the Roman empires before.

Austria, Germany, Poland, parts of "White Russia" etc, were all but parts of this. Where he was born in completely irrelevant to this (nor does it matter in the United States), and the concept of a natural-born citizen is not something Germany ever embraced in any way, shape, or form.

Not all nations have or had the rule about being a "natural born citizen". Hitler obtained German citizenship in exchange for fighting with Germany in WWI and apparently by the rules for German elections in that era he was eligible to run. 

cause they decided to do it for some reason

As I understand it, he lost the election, everyone thinks he won, but he obtained another position by appointment because of his political party's clout in Germany. He then forced his way in to the Chancellorship. Not like the Democrats who cheated their way in, but just as underhandedly. 

He was infatuated with Germany.  He considered himself more German than Austrian.  He was already living in Munich when the first world war broke out and wanted to fight for the Germans.  He appealed directly to the King of Bavaria (Bavaria was still an independent monarchy until 1918) to be allowed to fight for the German army despite being Austrian.  They intended to deny him but somehow the paperwork got mixed up and he was allowed to join the German army.  From there he went on to gain German citizenship and the rest is history.  All from an administrative error.