Playing seriously I would probably go through Turkey and enter the Soviets back door. Particularly the oil rich province of Baku.
Your back door approach to defeating the Reds sounds great but has some serious diplomatic, logistic, and military problems.
DIPLOMATICALLY, as Bulgaria is probably in the Axis alliance, you really are not "going through Turkey" but rather going thru your partner Bulgaria to first take Greece, and then going thru Turkey to finally attack the Soviets. This will result in all the provinces of Turkey and any further advance into Russia going to the credit of Bulgaria because - regardless of who annexes Greece (be it you or your partners Bulgaria or Italy) the Greek province of Kavala will go to Bulgaria because Bulgaria has cores on Kavala. Bulgaria will claim Kavala in spite of whatever advance you plan. In fact, Bulgaria will annex Greece - even if you don't - as soon as Athens is taken (and I am sure you will not omit taking Athens to eliminate a huge threat of remaining Greek troops on your flank). So, the only way around the fact that progress into Turkey will result in all provinces becoming Bulgarian is to use amphibs on Turkey to give Germany a new fresh landing past Kavala so future provinces taken are indeed German controlled and not Bulgarian controlled. If you try that (amphibing Turkey) you will encounter the next problem - LOGISTICS.
Even if you could place German transports in the Mediterranean to conduct an amphib on Turkey (only possible if you deny Italy the Yugoslavia provinces it claims along the Adriatic Sea by the way you conduct the conquest of Yugoslavia) your transports will not survive very long against the British fleets roaming the Med. But let's say you did manage to amphib Turkey with German troops to so start a new German controlled territory from which all further progress towards Russia results in the next conquered provinces also being German. Your supplies to those new German controlled provinces in Turkey and finally Russia will all run thru your ally Bulgaria and/or Italy. While that is not bad, unfortunately for you - once you get the oil rich prize of Baku - you will discover that you simply have no way to extract any oil because you don't have a single port you can convoy to. As both Gibraltar and the Suez are closed to you, and all other ports that you can reach are your allies (possibly Nationalist Spain, Italy, Bulgaria, Romania and maybe even Yugoslavia) you just have no way to extract any resources from that beachhead you won in Turkey and which you extended into Russia to reach oil rich Baku. In fact, you need to wait for your normal Barbarossa to conquer deep enough into Russia and get around the Black Sea to connect to your isolated territory in Turkey before you can extract resources. Likely you will also suffer serious TC problems for all your troops advancing past Turkey into Russia.
If the above two reasons were not enough to cancel any thoughts of attacking Russia via a German conquest of Turkey, perhaps the third reason will: MILITARY
Neither Greece nor Turkey are “push overs”. Assuming you are allied with Italy, Bulgaria, Romania, and Hungary you will still find that the military units you need to first take out Greece and then Turkey seriously deplete the strong position you could have against Russia if all those troops were on the German-Russian border facing Russia in a normal Barbarossa set up. The greatest loss of strength will not be in the obviously fewer ground troops you can position against Russia, but will be in the greatly reduced air force that you can put against Russia at the German-Russian border because you will need to relocate several of your Interceptor and some Tactical Bombers to the south to have success in the conquest of Greece and Turkey so you can march on to Baku. By the time you get those diverted air units placed into air bases you just conquered in Turkey you should loose terribly in any air encounter with Soviet planes because you are advancing onto bases that start at zero value and your aircraft are also suffering heavy loss of org due to rebasing forwards. It’s not at all like the 100% org planes on 100% airbases you can build along the German-Soviet border before Barbarossa starts. And the real killer of your German planes will be the fact that you are moving your weakest air units onto the worst airbases but taking on the very strongest part of the Soviet Air Force which is the two sets of interceptors that the Reds traditionally base at Odessa and Sevastopol and which will cut your air assets in Turkey to pieces. And without an effective German Air Force in conquered Turkey you will probably next realize just how easy it is for the Soviets to halt your planned advance to Baku by redeploying a few of their units to that most defendable mountain region of The Caucasus where you simply will not advance without adequate air support, mountain troops, and greatly superior numbers. But if you do, be sure to cover all the beaches along the way because any human Soviet player would be sure to build some transports at Sevastopol and land in your rear and then eliminate your whole “Baku Expedition Force”.
In my opinion, a German thrust into Turkey seriously decreases the chances of success for a German Barbarossa. Historically many people think that Hitler failed at Stalingrad only because he delayed the start of Barbarossa by three weeks helping out the Italians.
While the idea of a “diversion attack thru Turkey” is a good idea, it should not be done using German forces. It would be much better to let Italy realize it’s historical dreams of connecting from Venice to Tirana/Vlore (it has cores on all the Yugoslav provinces in-between) and possibly – if Italy smashes the British fleets in the Med – then let Italy amphib Turkey and take on the Soviet Black Sea Fleet. Italy can easily go around Greece if desired as they got the convoys and short convoy routes. In fact, Italy could go all the way to Baku. They could use the oil too.