James T. Kirk

In a parallel universe, James T. Kirk was the captain the USS Enterprise in the 23rd century.

On one occasion, the Eleventh Doctor recalled memories of his fourth incarnation meeting up with Kirk and the command crew of the USS Enterprise to combat a Cyberman invasion of Aprilia III. (COMIC: Assimilation²)

Other references
A fictional character named Captain Kirk appeared in the science fiction television series Star Trek in the Doctor's universe. (PROSE: The Gallifrey Chronicles)

Spock's Brain was an episode of Star Trek which both Izzy Sinclair and Destrii had seen. Destrii even went so far as to call it her "favourite." She cited "the bit where has to do the " as a particularly memorable moment and even quoted back a portion of the log, in unison with Izzy: "Captain Kirk's hunch that Spock's brain is on this planet appears to be correct!" (COMIC: Ophidius)

Donna Noble briefly adopted the alias of "Captain Kirk" to the Tenth Doctor's "Dr McCoy" when dealing with a group of soldiers. (AUDIO: Pest Control)

When DI Patricia Menzies entered the Sixth Doctor's TARDIS and saw its console room in 2010, she commented that she had expected the TARDIS interior to resemble "Captain Kirk's ship". The Doctor responded, "Detective Inspector, may I tell you something and I can't stress this enough: I am not Captain Kirk." (AUDIO: The Crimes of Thomas Brewster)

In 2015, Cory said he became an astronaut to be like Captain Kirk or Han Solo. (COMIC: The Hyperion Empire)

Behind the scenes
The mentions of Captain Kirk in The Gallifrey Chronicles, Ophidius and The Crimes of Thomas Brewster treat him as a fictional character (as do, implicitly, other references to Star Trek and Star Trek: The Next Generation in Doctor Who stories). Assimilation², of course, treats him as a "real" individual. It also shows that he exists only in another universe. It is thus possible that he is a character in the Doctor Who universe but a real man in the Star Trek universe.

Captain Kirk was briefly portrayed by Fifth Doctor actor Peter Davison in the closing sketch of an episode of The Harry Hill Show.