I'm am often seeing people writing about how Dean and Castiel are the same as Kirk and Spock, and most them them write how this justifies the Destiel ship. Now, I ship Destiel and am also a Star Trek TOS fan I felt I had to weigh into the debate.
I have analysed the three leading characters of both shows (Dean/Sam/Cas and Kirk/Spock/McCoy) and drawn parallels accordingly in relation to their characterisation and interaction with one another.
Dean as Bones
Dean I feel is more like Leonard McCoy than James T Kirk. Dean is the "old hand", the humanity. He is the one who will always put others before himself. He is the "healer" who, despite his best attempts, often finds himself emphasising with people. He is willing to sacrifice himself for the team, and will try to talk himself in and out of things.
Dean is old school, preferring to work with physical tools when he can rather than modern technology, although he is adept at computers. Again this is similar to Bones who, while he doesn't hate new technology, prefers the tried and true methods of hands on medicine. Bones and Dean also share a love of tradition.
Another similarity between Dean and Bones is their belief that they are responsible for their father's death. In both cases there was nothing they could do, but deep down they have never been able to shake the feeling of responsibility, and it is a guilt seeded so deep they will take it to their grave.
Castiel as Spock
I agree that Castiel is much like Spock. He is the other worlder, in a constant struggle between his "alien" side and his humanity. He is confused about emotions and tries to keep them suppressed, but his exposure to Sam and Dean often cause them to bubble to the surface and he eventually comes to accept them as a part of himself.
Castiel wants to do things according to how he is trained, falling back on his diverse knowledge and methods. However, he is not above following Dean's (or in Spock's case, McCoy's) lead in many circumstances, allowing himself to see the more human response and act upon it.
Like Spock, Castiel has also been in situations where the more human element of himself has been brought to the surface. We see this when Castiel encounters Famine, a "spell" making him crave meat in a way he views to be unbecoming of his angel self, but he can't keep it in check.
Sam as Kirk
If anyone is like James T Kirk, it is actually Sam. Sam is the more impulsive of the brothers, who wants to do what he thinks is right yet tries to stick by the books. He craves knowledge like Castiel/Spock, but also has a human side like Dean/McCoy. Yet somehow things often end up going wrong for him, especially with the ladies. He can handle his own in a fight, but he needs the other two to support him in order for him to be at his best.
Sam is able to adapt to any situation he is thrown into. He is a natural though ever reluctant leader, and can both negotiate and fight depending on the situation at hand. However, Sam's desire to do what he thinks is right has caused tension with both Dean and Cas, and often comes back to bite Sam.
Like Dean, Sam is willing to sacrifice himself for those he loves, as well as for the greater good. If he feels that his death will prevent the destruction of the world he will happily offer himself up. Like Kirk, Sam also has a deeply buried self-doubt that he only shares with Dean/Bones when they're alone, but in public Sam is the image of strength and self-worth. He will make others see the good in themselves, and will go out of his way to save those he views as innocent.
Team Free Will as The Triumvirate
In essence, Sam, Dean and Cas make up three parts of the whole just as Kirk, McCoy and Spock do. Dean and Cas balance each other out, but as often as they are perfectly in line with each other they are also at odds, and it is Sam who tries to smooth the tension between them.
Cas is often critical of Sam's impulsive nature and choices, which Dean will defend. When Sam or Dean need a shoulder to lean on they will turn to each other, connected on a deep level and happy to sit down and share their thoughts over a drink even though often words don't need to be said for them to know what the other is thinking or feelings, just like Kirk and McCoy.
Dean and Cas have a profound bond, much like the one shared by Spock and Bones after the events of "The Wrath of Khan" and "The Search for Spock" where Spock entrusts his katra (ie soul) to Bones, and the repercussions leave Spock and Bones with a permanent mental (ie "profound") bond.
At the same time Sam and Cas have a deep respect and trust for one another that has been forged over time. They have an understanding gained from experiences in insight, and value one another's opinions. Much of their relationship is based on their mutual love of knowledge and desire to learn. Sam wants to teach Cas about what it is to be human, and has significantly more patience with Cas than Dean does. Sam is also more forgiving of Cas' errors, even though Dean never hates Cas for his mistakes.
While each member of the three are more than capable of working on their own, they work best as a team. This is as true of both Sam/Dean/Cas as it is Kirk/McCoy/Spock, as evidenced on their respective shows/films. Individually they have weaknesses the other two can help balance out, and they are more likely to succeed as a team than alone.