Since you must have Battle for the Bosporus for this achievement, it makes restoring Byzantium easy. At the start of the game, arrest the king, then go down the democratic path of the Republican tree. Make concessions to the EEE, as you'll have to flip to fascist to form Byzantium and these concessions will mean a stronger fascist base when you do flip. Eventually, you'll be able to start the Heraklion Convention. The conference must fail for the fascist takeover. Success means the Megali Idea, which is a separate achievement entirely. Failure can come in multiple ways, but the easiest is to play the game with historical focuses on and invite all Sevres signatories (UK, France and Italy) to the Heraklion Convention. If Italy is invited, France and Britain are guaranteed to refuse to go unless they themselves are also fascist, and if they don't come, the conference is cancelled. The next focus, Horror and Fear, will immediately put you at war with Turkey and, unless they've cancelled their guarantee, Romania.
With that said, the rest of this will rely on your skill and luck, as you'll have to face both Italy and the USSR, both much more powerful than you. Allying with the Germans can help with the USSR, but your main problem is the Italians, who will generally employ naval invasions which you do not have the manpower or fleet to fight against. And, unless you can outproduce Germany and usurp control of the Axis from them, you're going to have to face the Reich when you finally do take on the Italians. Some words of advice, though you don't necessarily need to follow them:
> After defeating Romania, take their southern and coastal territories for yourself before you puppet them. You'll need these territories for the "Triumph in the Balkans" decision, which will give you more core territory, meaning more manpower and factories to work with.
> Puppet USSR and use their manpower to bolster your army. Even if you just use basic Russian infantry to conserve your limited factories, the USSR has several million manpower from the start of the game, compared to your barely 1 million if you're lucky following the conquest of Turkey and Byzantine Restoration, and if you're smart with your takeover, most of that manpower should be your puppet's.
> Just because Germany is a potential ally, doesn't mean you should help them. Whilst you will need to fight the Allies if you decide to reform the Byzantine Empire in it's entirety, the achievement only requires the initial reformation (Greece and Turkey) and the three target nations puppeted. Use Germany to take out the USSR, then either leave the faction and try to join Japan (using the same rule - no helping against the USA) or declare your own war on Italy when the Allies start pushing the Axis back. As I said before, it's going to require luck, but having the UK, France, America and maybe Japan on your side, even if they're all distracted by their own wars, will still take some of the pressure off you.
> With the 12.0 update and By Blood Alone DLC's release, the peace system now allows you to take a nation's navy. After dealing with the the Soviet Union, claim as many of their ships as possible, to help bolster your navy against the Italians. This might be a reason to ignore the previous piece of advice and join the war against the Allies, as you can then take Britain, France and America's navies too, assuming the AI doesn't beat you to it. Just make sure you puppet the USSR before taking ships, unless you want the Germans to take all the land from you and leave you having to fight them as well.