What was Nazi Germany's attitude to the former German Royal Family?

Hitler and the Nazis took a very dim view of the German monarchy and were quite hostile to monarchist sentiment and supporters. That being said, he did allow the younger generation of nobles to serve as officers, seeing that they were more open to national socialism than their predecessors and seeing that they were invaluable as officers. Many of Kaiser Wilhelm's sons had children serving in the army, and many royals died in service to the Wehrmacht. But most royals were kicked out of the army as the war went on and Hitler grew increasingly paranoid about a possible monarchist coup. He held Bismarck in quite high esteem, and he saw the Monarch, specifically Kaiser Wilhelm as being the main stumbling block to Germany's success. This quote from Hitler during a meeting in 1941 is quite illuminating:

The injustice committed by the Kaiser at Bismarck's expense finally recoiled upon him. How could the Kaiser demand loyalty from his subjects when he had treated the founder of the Reich with such ingratitude? The shameful thing is that the German people allowed such an injustice to be committed. The generation of 1900 was lost—economically, politically and culturally.

He also spoke critically of the Kaiser's ability to command troops (with no sense of irony):

Apart from the great victories, like the battle of Tannenberg and the battle of the Masurian Marshes, the Imperial High Command proved itself inadequate. The Kaiser put in an appearance on one single occasion, because he believed that all would go well. During the great offensive of 1918, it was trumpeted around that the Kaiser was commanding it in person. The truth was, the Kaiser had no notion of command.

Hitler, at a dinner party in May of 1942, again went on a long rant about the Kaiser:

The behaviour of Wilhelm II in society was unworthy of a monarch. Not only did he consistently ridicule the members of his immediate entourage, but also fired a constant stream of ironic remarks at his guests for the amusement of the remainder. His bad taste and familiarity with other monarchs—backslapping and the like—robbed Germany of much sympathy. A monarch must learn that self-restraint and dignity must be observed in everyday life. The example of Wilhelm II shows how one bad monarch can destroy a dynasty. In the same way, those who wish to play their parts in history must understand that one single bad generation can cause the ruin of a whole people.

Now Hitler didn't hate monarchy as rule of thumb, indeed there were many German Kaisers and monarchs he respected, but he held dim a view of the Kaiser Wilhelm's monarchy.

Himmler had a different view of the nobility, while Hitler saw them as needlessly decadent and incompetent. Himmler saw their years of selective breeding as being a good thing, assuming it would produce great racial qualities. As a result many young nobles joined the SS. By 1938 a fifth of the top SS officers were nobility. Many like Erich von dem Bach-Zelewski (who played a key role in the suppression of the Warsaw Uprising) found a home in the upper ranks of the SS.

So moving on from the military, things were not good for monarchists on the home front. In 1934 the Brownshirts and the Gestapo began cracking down on displays in favour of the monarchy. For example private celebrations of the Kaiser's birthday were banned, and celebrations were ruthlessly broken up. Hitler lavished bribes upon old monarchist officers like August Von Mackensen in order to drive them away from Monarchist sentiments. This particular case caused a bit of a ruckus because Von Mackensen wrote to the Kaiser asking if it was okay to accept a gift from Hitler. This show of respect managed to anger both Hitler and the former Kaiser. Finally, with the death of Hindenburg and Hitler's ascension to absolute power, many Monarchist organizations were banned and any hope of a restoration was gone completely.

Among the monarchists opinion was divided over the Nazis. During the Wiemar years a clear generational gap had emerged. During the Wiemar years many aristocrats had been deprived of their estates and had grown impoverished and had been politically marginalized by the Republic. They longed for the return of the good ole days of the German Empire. To the monarchists the Nazis plans of conquering and colonizing the eastern territories appealed to them and reminded them of their German ancestors who had done the same in Pomerania and Prussia. But to the older monarchists were also familiar with extremist groups. They viewed the Nazis with distrust and saw their fanaticism as being a fatal flaw. The Nazis racial speeches and violent tactics did little to win over the old monarchists.

However, the generational gap had created a group of young, radicalized nobles. These nobles, many of whom belonged to the former royal family held much the same views as Hitler with regards to race and foreign policy. Much like Hitler, these young nobles despised the older generation for surrendering in 1918. This group of young nobles supported the Nazis. And the Nazis made great use of these supporters during election time, using them to win votes from conservatives who otherwise would have been put off by the Nazis fanaticism. Prince August Wilhelm of Prussia and Crown Prince Friedrich Wilhelm were both early supporters of Hitler.

So how did the Kaiser himself react to Nazism? Well he held much the same view that the older generation did. He initially supported the Nazis assuming that they would restore the monarchy, but once the Nazis made it clear that wasn't going to happen, he turned against them. He viewed things such as the night of the long knives, where the Nazis purged many monarchist supporters among others, as "gangster tactics" that had no place in Germany.

Kaiser Wilhelm in his later years kinda waned between being a supporter of Nazi racial doctrine, and being against it. He would often complain of a Jewish plot against Germany, with Jews manipulating Britain against Germany. But he also opposed the violence of things like the Night of Broken Glass, where Nazi supporters and angry German citizens attacked Jewish shops and synagogues. He said "What would people have said if I did such a thing?"

There was an incident before WWI, where Kaiser Wilhelm's son was given a book, written by a nationalist author named Konstantin von Gebsattel, called "If I were Kaiser" which argued for harsh measures against the Jews and the Social Democrats; as well as authoritarian rule by the Kaiser. The author forwarded his ideas to Kaiser Wilhelm's son, a known nationalist. The Kaiser shot down his son's and Gebsattel's ideas as "Childish" and "potentially destabilizing." Furthermore Gebsattel was regarded as a "Weird Enthusiast." None the less the Kaiser did read the book "Foundations of the Nineteenth Century" by Houston Stewart Chamberlain and looked fondly on its anti-Semitic stereotypes. But the Kaiser never dreamed of actually acting on those ideas, and didn't dream of seriosu restrictions on the Jewish population.

When the Nazis took over the Netherlands, Kaiser Wilhelm was given a permanent SS guard and people were forbidden from visiting him to pay respects. Even Hermann Goering, the number 2 man in Nazi Germany, couldn't visit the Kaiser. Which speaks to Hitler's irrational fear of Monarchist sentiment.

Edit:

For the quotes I drew directly from Hitler's Table Talk and Mein Kampf. For the Nazis smashing of the Monarchist movement I recommend Richard J. Evans' treatment of the Nazi political movement in his book "The Third Reich in Power."