I'm pretty sure this is it.
According to the description is was recorded in 1996 by Kevin Barbare. I don't know how to answer without just including a link.Here's a direct link to the video on Youtube if it doesn't play here.
Kirk: I am the captain of the Starship Enterprise.
Spock: Captain Kirk: This is Spock.
Please step on the transporter.
Scotty: Dilithium crystals!
Hit by Klingon missiles!
No!
McCoy: I'm a doctor!
Kirk: Not an actor!
McCoy: Not a milkman!
Kirk: What does that mean?
McCoy: And I'm sorry -- He's dead, Jim.
Chekov: Photon torpedo . . . !
Kirk: Lieutenant Uhura,
Open hailing frequencies
Uhura: Yes, Captain Kirk,
Opening hailing frequencies, sir
Kirk: Let's boldly go where no one's gone before!
Ha ha ha ha ha ha . . .
Beam me up, Mr. Scott;
Sulu, go to warp!
Sulu: Warp three, sir.
Kirk: No! That will be way too slow!
Sulu: Warp four, sir?
Kirk: That still is way too slow!
Sulu: Warp five, sir.
Kirk: It still is too damn slow!
Sulu: Warp six.
Kirk: It's too damn slow!
Sulu: Warp seven?
Kirk: It's still damn slow!
Chorus: It's too damn slow . . . !
Kirk: No, no, no, no, no, no, no!
Mr. Chekov, Mr. Chekov --
Chekov: I am firing torpedo!
Spock: That will not work and would be illogical
To me, to me, to me . . . !
Some sources credit Weird Al and others claim it was originally created on something called "The Hill-Man Morning Show".
