Some of the moves most frequently used by William Shatner in the role of James T. Kirk included the double-clenched hammer blow, the body roll into one or several attacker's legs, and the infamous "drop kick."
Ironically enough, the vast majority of "Kirk-Fu" fighting techniques actually have a basis in "real" or at least, traditional martial arts.
The aformentioned clenched hammer blow can be found in Tae Kwon Do, while the stomach throw (where Kirk would place his foot into an opponent's gut, roll back on the floor, and send his opponent overhead is from judo ("tomoe nage").
Even Kirk's version of the roundhouse kick resembles a Thai kick delivered with the shin, as used in both "Arena" against the Gorn, and the infamous "Spock's Brain", where Kirk utilizes the roundhouse kick against one of the male guards holding he, McCoy, & Scotty prisoner.
It is important to note, however, that though Kirk utilized the shin to deliver his version of the round kick, Muay Thai was not especially popular in the United States in the 1960's. It's much more likely that William Shatner, having trained in karate, used his own, adapted version of the roundhouse.
On that note, it would seem that the vast majority of Kirk Fu techniques come from either karate (the shuto "chop", used in both "Amok Time" and "The Omega Glory", being a prime example) or judo (Kirk actually gives "Charlie X" a judo lesson in the Enterprise gymnasium.)
It's only natural that judo would make up the bulk of Kirk-Fu's inspiration, since judo was arguably the most popular Asian martial-art in the United States during the 1960's, and "Kirk Fu"-like judo techniques can be seen in other shows such as "Wild Wild West", "Honey West", "Mission Impossible", and of course, the James Bond films.
Due to its extravagant nature, Kirk-fu is often mocked and occasionally parodied.