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 producers, writers, and director all felt that this was a comic superhero film. (Hence the appeared of several issues of ROM space knight, a forgettable Marvel book from the 80’s.)
with latex to make him look merged with the machine, the image is disturbing. We can emotionally feel Murphy’s loss of humanity. When he says he can feel his lost family but he can not remember them, it strikes home as true.
the suit.
does it mean to be devout, it found itself measured as product and summarily sentenced to butchering before release. Tossed out to die an ignoble and forgotten death then film slowly built a following. From the strange images, the non-cinematic score, and the brutal inescapable ending, the film became legend. Interest grew, interested in perhaps the director’s original vision, not subject to an executive’s callous command to cut fifteen minutes and he didn’t care which. The birth of conspiracy, when it was discovered that all the original negatives had somehow inadvertently been used as landfill in building a highway. All of this merged into a strange and almost unbelievable history for a simple low budget horror film.
simply it never was around in a form for me to watch at the time and place where I had an interest. I had seen scenes and I had even seen the closer couple of shoots, but never the entire film, and certainly never in one go from front to back. It is an interesting experience, particularly from a position 40 years after it was released, and after it had spawned a franchise of its own.