In the beginning, Barbarossa was launched with the assumption that the Soviet Union would collapse quickly. This did not happen, for various military, economic and political reasons. On this basis, the Germans attempted to convince the Japanese to strike at Great Britain at places like Singapore and Hong Kong late in 1941 and facilitate their war against Britain. They kept the Japanese in the dark about Barbarossa.
Amidst the signs of obvious failure just short of victory in Russia, the Germans tried to convince the Japanese in October and November of 1941 to attack the Soviets from Manchuria. This failed, as the Japanese already had plans to invade the Philippines, Singapore, Borneo, and all associated islands to secure oil supplies. Moreover, they had concluded a non-aggression pact with the Soviets they preferred to keep in place. So the Soviets ended up at war with just Germany and Italy. The Japanese ended up at war with just Britain and the US. The Germans, however, ended up at war with all three major Allied powers, by foolishly declaring war on the US just after Pearl Harbor.
The German DoW was predicated on a weak American response to Europe based on the Japanese threat. The agreement to focus on Germany between the Anglo-American powers confounded this idea. Even Mussolini believed that the focus would be on the Pacific after the Japanese attack, hence his declaration of war on the US, which seems insane considering the results just over a year later.
At that point, Hitler's (and Goebbels', judging by his diaries) only hope was for a cleavage between the Soviets and the Anglo-American powers. There was about zero chance of this happening with Roosevelt championing the alliance, but that was what he hoped for. When Roosevelt died and nothing changed, this is the point where he gave up hope and we can then segue to the events of "Downfall".
I don't think Hitler hoped for actual military victory post-Stalingrad. He hoped for delay in the war's end game to present a political situation he could take advantage of. His thoughts were rational, I believe, but unlikely from this vantage point to ever bear fruit.