As the title informs you I will only be speaking about the Klingons as they appeared in the original Star Trek series, not he later retconed aliens that were introduced in the Motion Picture and later elements of the franchise.
Star Trek arrived during an interesting period in entertainment history, by being produced in the later 60’s the series was influenced by and could be seen as a metaphor for both World War II and The Cold War. The writers, directors and producers of the series 
counted among their number several World War II veterans while the Cold War, now well under way, underpinned everyday life and provided an atmosphere of dread under the treat of global nuclear war. It is natural for people to look on the Klingons of the original series as a metaphor for either the NAZIs or the Communists, but which is a better fit?
Now first I am not putting forth a proposition as to what was in the creators heads when they crafted Star Trek’s original bad guys, but rather just an exercise about which brutal ideology best matches what we know of the Klingons.
There are of course a number of areas of overlap between the deadly forces of Communism and National Socialism. Both were brutal dictatorships, both murdered on a 
vast scale, both were ruthlessly expansionistic, both were extremely militaristic, both engaged in suppression of dissidents, both crushed the individual under the power of the state.
All of these elements would seem to apply to the Klingon Empire, both this complicates the issues. However it is important to recognize that National Socialism and Communism however similar in many area are not the same thing, so perhaps by finding key differences between the two and apply them to what we know or can extrapolate from the Klingons we can determine the best match.
Communism did not recognize the right to own property. All property belonged to the state, while Nazism recognized private property in a fairly recognizable state, corporations, 
and the like. From the original series we have no data about the Klingon economic models and systems, so we can’t use this as a point of differentiation.
Communism believed in an inevitable march of history; that peoples and cultures inexorably moved through certain developmental stages, in the same sequence, and that the end results would be the stateless commune. Nazism believed in the survival of the fittest on both a racial and a cultural, which were really one and the same for them, stage. They thought that it was the natural order for the strongest culture to dominate and subjugate the ‘weaker.’
Ah here we have a bit of a match with the original series Klingons. They clearly believed that if they were stronger is was natural and right for they to rule. They did not argue their dominance from destiny, but from ability.
Communism was a very delusion and distorted view of mass teamwork. That everyone person, if given an equal share would pull equally hard for the greater good and that want and greed would die away. It was group oriented, but in a fanciful belief that people would become happy and prosperous in a share and share-alike fantasy. Nazism saw the individual as only a cog in a machine to support the state, and the race which defined the state. Every man and woman had a duty to the state and that duty overrode all individual consideration. There was no utopian fantasy of universal brotherhood, only the importance of the state over the individual. This matches up quite nicely with Commander Kor’s speech to Kirk in ‘Mission of Mercy’ just before his men burst in to, once again, arrest the Captain.
The final point of divergence is the spread of ideology.
The communists could be though of a evangelicals. It was not enough to conquer territory, to claim resources, to amass power, they also had a burning need to convert. They desires that not only did subjugate peoples bend knees to their power, but adopted their view and 
vision of the future. Like the Inquisition before them the Communists could brook no heretics. The Nazis weren’t interested in that at all. Because of the racial components of their belief system, they saw the world as a conflict between themselves and everyone else. They never expressed an argument that their system was better for everyone, they never tried to convince Kirk or the Organians that they should convert, but rather brute force to take was enough and all they required was subservience.
To me this makes it clear, for the Original Series Star Trek, the Klingons were much more like Nazis than communists.
The Fascists Were far less race oriented than the Nazis. In fact until their subservience to Hitler’s Nazism, there were many prominent Jews in italian Fascism . Fascism was much more a nationalistic movement than a coherent ideology. Even though it predates the Nazi’s rise to power by a decade, in most people’s minds Italian fascism is often merged with german Nazism.
The Japanese were incredibly race conscious and in many ways the worst racists of the war. On could make the argument that the Romulans in many ways were Japanese metaphors.
I agree. Communism, even the Stalin/Leninist variety, was not as race conscious. As you said, the evangelical urge to convert. However National Socialism, it cousins fascism and the military version in Japan were all extremely race conscious and this pervaded all their actions. In this aspect, Klingons were much more Nazis.