There are no trains from Greece to any other country right now.

You'd need to go by bus at least partially.

On the most direct route via Belgrade, you'd better go by bus completely. Unfortunately, here are no trains from Greece to North Macedonia, no trains from North Macedonia to Serbia and no trains from Serbia to Hungary. 

It's possible to go by train from Bulgaria to Romania and from there to Budapest. Long journey and there's still a bus needed to go to Sofia.

I don't understand then why the interail site allows us to make an itnerary including Greece to Budapest. Later it says invalid but it says invalid to every train trip I included in my designed itinerary (although I know there are operating trains, eg Prague to Vienna). Are there any specific tricks to design an itinerary that is doable? 

You might take a bus from Saloniki to Sofia and from there go by train (via Romania). The bus is not included in interrail but shouldn’t be too expensive (around 20-25€ if prices didn’t increase in the last years)

@Hugo V You might want to take a look at this.