No. The coup he did was illegal de facto, since he tried to seize power through illegal means (not through elections). Then, as to question whether everything he did as the Fuhrer was legal, that's more of a philosophical question. What the Nazis did was legal under the Third Reich's regime, but that's easy when you are the one that defines legality.

To answer your question : no, everything he did wasn't legal. It became so when he became the Fuhrer, I guess MLK was trying to explain that legality is in the end of the people in power; if it's a racist dictator, then racism can be legal, even though it's a terrible situation. A more accurate way to present it would be to say that everything that the Nazi Government did was legal.