It is important to emphasize that the war was necessary for the Holocaust, essential even.

The Holocaust in the form that it occurred was not planned that way in 1933 and not even in 1939. For the Nazis the so-called Final Solution to the Jewish Question took several forms depending on when you ask. In the 1930s the preferred and conducted policy was forcible emigration from Germany to basically every country willing to accept German Jews.

Accepting the ideological delusion that the Nazis were fighting a war with the Jews, it was actually their own success in the war that placed them under Zugzwang. For the Nazis Jews were a security concern and every Jew under their rule was an additional security risk. Therefore, occupying Poland with its approx. 3 million Jews really put the pressure on the Nazi leadership to find a solution to this purported problem. The Madagascar plan is a perfect example for this.

Additionally, the war was important for the acceptance of murder as a policy. With the progression of the war, the Nazis saw a chance to have a united German populace against an external enemy in order to deal with "internal enemies". Thus, with the start of the war, the euthanasia program is initiated. That was an important step towards more murderous policy because it proved to the Nazi leadership that it is indeed possible to just kill people.

The most important step probably is the attack on the Soviet Union. The war against the Soviets was from its very inception in the minds of the Nazis a war of ideological annihilation aimed at complete destruction of the "Judeo-Bolsheviks". Therefore, when the attack on the Soviet Union begins, the Kommissarorder and other similar orders aim at murdering the Jews of the Soviet Union because they are for the Nazis the same as a member of the Communist Party. And out of this decision grows the policy of murdering all Jews of Europe.

In fact, it is likely that the intial good outlook and then especially the not-so-good outlook in how the war went, was an important factor in speeding up the killing process. In the minds of the Nazis, the weaker they were, the greater the risk from the Jews. And so, to them, it was necessary to kill them even faster.

You mustn't see the effort put in the Holocaust and the war effort as something separate or contrary. For the Nazis, this was the same war they were fighting and the effort for one had the same priority as the other.

Sources:

• Richard Evan's Third Reich Triology

• Raul Hilberg: The Destruction of the European Jews, 2003.

• Christopher Browning: Fateful Months.

• Ian Kershaw: Hitler.

• Ian Kershaw: The Nazi Dictatorship. Problems and Perspectives of Interpretation, (London, 1985, 4th ed., 2000).

Commiespaceinvader basically summed it up really well, but I'd like to add one point.

By November of 1941, as per Lukacs, Adolf Hitler had realized that he could no longer decisively "win" the war in his way, contrary to common historical memory. He might not have shouted it from the top of the Berghof, but he knew it, given his interactions with Todt and others at this time. Hitler was a consummate actor: how he behaved always depended on his intentions around the person, and that shows with the public face of optimism contrasting with more revealing private recollections. Jodl at Nuremberg, among others, testified that it was wishful thinking to say that somebody should have told Hitler that the war was "lost" given that he knew it before anybody, and I agree.

(What he did after 1941 was shift to a Frederick the Great style strategy of trying to outlast his opponents and hoping that the unnatural coalition between capitalists and Communists would break up if he fought long and hard enough. That eventually occurred, but he fundamentally underestimated the will of the enemy to see him gone before that time, particularly in the West. Such a strategy thankfully went against his personal strengths, which largely were contingent on him being on the offensive and taking the initiative-he was completely at a loss for defensive warfare. Among other things, this caused his increasingly poor decisions from late 1942 onward.)

Even since Barbarossa started, the Einsatzgruppen had been killing Jews by the thousands in wild mass shootings, but the plan for European Jews was to cart them off eventually and have them labor for the Reich. That was no longer feasible with the prospects of a rapid military victory of the Soviet Union all but gone by that time.

I think it is not a coincidence that it was around this time that the official Final Solution came about. The Wannsee Conference gets more attention, but the true evidence of the Final Solution lies in the December meeting at the Reich Chancellery in central Berlin, which came right after Pearl Harbor. That meeting involved Hitler and the bigwigs rather than the bureaucrats who would implement the plans, and Goebbels and Frank made it quite clear what Hitler had in mind in broad terms for the Jews given the new character of the war. The original plan to expel the Jews into the wilds of Russia was not going to occur, they no longer had any value as hostages and were taking up precious food resources. And as the war went on and things got more and more grim, Hitler saw it as a personal war he could "win". It had a very twisted internal "logic" to it. If you believed the stab in the back myth, then as per Himmler at Posen in late 1943, it was thanks to the Jews being gone that there was no grumbling on the home front like there was in 1917.

The Fuhrer was not an irrational madman, or at least not in 1941 yet. He was a cold, calculating, sociopathic mass murderer, following logic. His own twisted logic, but logic nonetheless. That's a very important distinction when it comes to understanding why what happened happened.

In addition to the excellent comments here, I'd like to add a point Timothy Snyder has argued: The war was a necessary condition for carrying out mass extermination.

The SS destroyed governments in Eastern Europe, including the Baltic States, Poland, Belarus/BSSR, and UkraineSSR. ~1% of Jews survived in those regions.

"Where states were destroyed, Jews were murdered; where the state remained intact, Jews could find some protection in bureaucracies and passports. It was in the stateless regions of Eastern Europe where the Nazis were able to experiment with and calibrate the Final Solution, which they then tried to export back west." (Edward Delman reviewing Snyder)

"In Nazi-occupied but technically sovereign Denmark, 99 percent of Jews survived, whereas in Greece and the Netherlands—where a puppet government ruled and where the SS took over domestic policy, respectively—three-quarters of Jews were deported to extermination camps. Only through the production of statelessness external to Germany could Adolf Hitler’s “biological anarchist” worldview find expression.

"Critics may object that it was precisely a powerful German state that created these stateless zones, but German Jews were still far more likely to survive than Jews in zones of statelessness. Further, the chief murdering institution—the SS—was a Nazi Party organization whose ability to destroy states multiplied once it left the borders of the Reich." (Timothy Nunan reviewing Snyder)