Invading Russia wasn't a strategic mistake at all; it was a (arguably the) primary objective. Hitler wrote about colonizing the East as far back as the 1920s, calling the Volga river Germany's Mississipi.
Hitler made a large number of strategic blunders, but here are a few in condensed form that better historians can explain or contradict.
First: Hitler shifted focuses during Barbarossa; IIRC he focused first on taking Moscow and then shifted to the Caucasus mountain oil fields. This cost priceless time and energy as units had to redeploy and such. Moscow is the transport hub of the USSR, plus more than 10% of Russia's population currently lives there. A brief looksee on transit maps of the USSR supports the old adage that all roads in Russia lead to Moscow. Loss of Moscow would have been devastating, a possible war-winner for Germany. Taking Moscow would greatly impede supply, rearmament, reinforcement, production; Moscow was equivalent to NYC and DC combined in terms of national identity. Its loss would have been a major morale blow to the Russians. The Caucasus were a leading oil supply to the Soviets before lend-lease really took off. The loss of either the Caucasus or Moscow would have been devastating. If even half of the losses of the Eastern Front are mitigated, it is entirely possible that Germany manages to hold off the Western Allies, thanks to the large amounts of raw materials, slave labor and industrial centers captured in the East.
Secondly: Hitler refused, against almost all counsel, to allow units to pull back from Stalingrad. This cost him an entire army and gave the Soviets a major morale and propoganda boost.
Third: Hitler fortified his positions in the east, disallowing retreat in favor of static defense (fortress cities). The strength of the German army was wasted in this form of defense, allowing encirclements.
Fourth: During the Blitz, the RAF did a retaliatory (spell that how you like) strike on IIRC Hamburg. The Luftwaffe was focusing on British airfields at the time, and was doing quite a number on the RAF. After the bombing, Hitler ordered the Luftwaffe to switch to terror bombings of English cities. This gave the RAF extremely valuable time to reinforce and rearm. It is speculated that if Hitler didn't intervene against airfield strikes the RAF would have been rendered impotent.
Fifth: Hitler supported Mussolini in Greece, pushing back the invasion date of Barbarossa by IIRC a month. That may not sound like much but in the context of the countdown to winter 1941 every second counted. An extra month of summer would have been a powerful resource in 1941. I don't know enough about Eastern European weather to confirm, but IIRC the Rasputitsa is over by May.
Six: Hitler treated Soviet civilians and POWs terribly. Huge numbers of them would have been willing to fight for Germany ( a lot of them did, see Vlaslov's Army) if they had not been brutalized by the Nazi armed forces. Initially, many considered Barbarossa a liberation rather than an occupation. That attitude didn't last.
(TL: DR)All of his fuhrer faults find themselves germinating from a consistent underestimation of "lesser people."
The biggest mistake Hitler made was actually his world view.
Viewing Slavic peoples as sub human would prove to be a horrible Achilles heel to the German war machine. German High Command strategies assumed racial true-isms of the Nazi ideology. When these plans dealt with reality, they were warped and always failed to react properly because of the "inferior race" assumption.
A good example of this is the entire offensive for 1943. Even though Hitler did not directly plan this offensive, his polices and politics were used to break all of the rules of Blitzkrieg. Remember the concept of "lightning war" requires timing, concentration of overwhelming forces, and surprise to be effective.
Operation Citadel came in a place both sides knew likely (Kursk), the German army attacked with less men then the defenders(~900K Ger, ~1.900 Soviet), with less mobility (~2,900 Tanks, ~5,000 Tanks), and even included delays. The Soviets had time to plan for the attack and create an amazing defense in depth.
The reason the High Command thought this would work despite all of the problems before they even began to attack...They were Germans fighting a Blitzkrieg against inferior people.