How do I access the hidden mini game in Call of Duty: Black Ops?

I believe you are referring to Dead Ops Arcade. Here's an explanation I found of how to access it.

Accessing it isn't just a matter of finishing the game. You've got to bust your way out of the main menu screen, then work your way around to a terminal and type in the correct commands to boot the game.

While in the menu look down (using the right analogue stick, presumably) at your hands and mash the triggers to wriggle your hands until they come free.

You'll stand up and be given the freedom to roam. Go to the terminal, access it and enter CD, then DOA. Bam - giant machine guns, flame throwers and rocket-firing bats are yours for the shooting.

Yea just hit both yours triggers or r2 and l2 buttons at the same time, go to the computer behind the chair you are sitting in, then type in doa on it. There's a tutorial here: http://www.Youtube.com/cjonduty

In the PC version, you have to hold the SPACE button in the main menu to free yourself and walk around. Then you can go to terminal and explore what it has to offer.

One thing I found really funny was the option to sit down - why should one give in willingly to be tied and beaten up? But that is necessary to access the main menu again.