Tolis Malakos of London Metropolitan University wrote a very insightful piece on this in 2013: What does the rise of fascism mean for Greece and for Europe?
Above and beyond that, turning to populist, authoritarian solutions when faith is shaken in bourgeois democratic politics is not an idiosyncrasy of Germans or Italians: it is human nature. Golden Dawn is already running charities for the poor—as long as they are of pure Hellenic blood.
Can fascism grow? It already has. Golden Dawn (political party) – Wikipedia: 0.1% in 1996, 7% in 2012.
Can it prevail, and do away with the current democratic system? That’s a different question. From my very remote perspective, there’s a vacuum of credible alternatives in Greece right now, and Greeks have gotten used to having their backs to the wall for the better part of a decade. It’s not the most likely outcome; but it’s not as inconceivable as it was two decades ago, when Golden Dawn was polling at 0.1%.