How did Hitler gain absolute power?

Though Adolph Hitler served in the German Army during the Great War and though he entered German politics under the auspices of the German Army's political wing, he remained an Austrian citizen till April 1925 when he renounced his citizenship so as to avoid deportation after completing his prison sentence for his participation in the Munich putsch. 

Hitler became a naturalized German citizen only on February 25, 1932. This was done by his being appointed to a minor civil service role in the state of Brunswick (where his party had won the  election). This automatically conferred citizenship on him and allowed him to run for president. One year later, Hiter became Chancellor and then, after the death of President Hindenburg, he became Fuhrer.

What explains this dizzying ascent?

The answer is that at every step of the way, Hitler was pushed forward by the Army whose maximal program involved looting reparations from France while gaining agricultural land and other resources from territory to its East.

 Suppose Ludendorrf hadn't been utterly mad (he was as anti-Catholic as he was anti-Jewish) and suppose he hadn't broken with Hindenburg, then he would have been Hindenburg's Chancellor. As things were General Schliecher got that position but he alienated Hindenburg by seeking to reduce aid to Junker landlords in the East. Moreover, General Blomberg hated him because Schleicher had sidelined him by sending him to East Prussia in 1929. 

In early June 1934, Hindeburg gave Hitler an ultimatum. Either he crushed the SA under Rohm (which was becoming a rival to the Army) or he himself would be stripped of the Chancellorship. The Army would rule directly. Bruning, a former Chancellor, was tipped off, and fled Germany on June 3 1934. One way or another, there was going to be a big blood-letting.

Hitler's 'night of the long knives' in late June was a great success. He didn't just personally arrest Rohm but ordered him killed. Rohm didn't see this coming. Hitler had brought him back to take over from Stenner, the previous rebel leader of the SA. But Stenner had been permitted to go into exile). More signinficantly, he had General Schleicher and his wife shot in their own home. Bredow, Schleicher's sidekick, too, was killed. This was the Führerprinzip in action. Here was the Caesar who would have no truck with either Democracy or party faction. Bloomberg was delighted and got the army to take an oath of obedience to Hitler on August 2- the day Hindenburg died. After that, Hitler assumed absolute power. He then proceeded to fulfil the Army's program better than they could do so themselves. He also got rid of Blomberg on the grounds that the fellow had married a prostitute. By the time senior officers tried to move against Hitler, it was too late. They were strung up with piano-wire. 

Hilter gained absolute power because he delivered what the Army wanted and the German people, misled by stupid economists, believed that only the Army could save them from starvation. Sadly, because the Army was as stupid as shit, the outcome was the utter annihilation of the German Army. The Commies were delighted because they got to rule East Germany. The West chafed under Allied occupation- Adenauer wanted to get nukes!- but then the economy grew at a miraculous rate. The Germans finally understood that Karl Ballod & Keynes etc. were wrong. They wouldn't starve if they failed to acquire land to their East. Rather thay would become more and more prosperous selling manufactured goods and importing as much food and raw materials as they wanted. They could have a 'hard' currency without having to get French gold (which is what had enabled them to go on to the Gold Standard in 1871.

Suppose a General like Schleicher had succeeded Hindenburg. Would the subsequent trajectory of events been more favourable for the country? 

I doubt it. The fact is German generals quarelled with each other. The chain of command was weaker. Some say this was because of its Auftragstaktik (mission command) leadership philosophy which prioritized decentralized decision-making. Thus the Army needed a Kaiser like commander or else its esprit de corps migght be fatally compromised. Equally, the Kapp putsch had shown that 'Civil Society' was loyal to the Head of State. Only if the Kaiser, or President, delegated absolute power to the High Command would the orders of military officers be obeyed. In other words, militarism wasn't innate in the German population. Ceasareanism was. Sadly, Hugo Preuss & Max Weber gave Weimar a constitution which allowed for a Caesarean President. Hindenburg was too senile to do very much after divisions within the SD fold caused the Legislature to accept rule by Presidential decree. Hitler was younger- he became Fuhrer at the age of 1945- and he was fortunate in that the worst of the Depression was over. The Communist movement too was fragmenting or otherwise losing momentum. People could believe that a lurch to the lunatic fringe of the Right would be reversed once traditional sources of authority- the Church, the liberal professions, the more cultured of the industrialists and financiers- were able to reassert themselves. 

Could Hitler have risen to power if there had been no Great Depression? Yes. Look at Mussolini. The King himself appointed him (he also dismissed him after it had become clear that the Axis had lost the war) during the Roaring Twenties. 

What changed for Germany was the end of 'extend and pretend' which would have happened in any case even without the Stock Market crash. Once there was no net inflow on the capital account, the German Army no longer had an incentive to do secretly what they would soon do very openly.