I'm trying to hook up my old United UK-8000 Karaoke machine to a
system consisting of a TV and a DVD / VCR combo unit.  Currently, when
I play the karaoke CDs, I see the words and hear the music from the CD
and see the words on the TV.  I am currently running the RCA audio out lines
from the karaoke machine into the DVD / VCR unit.  The signal then
goes to the TV set.

What I don't hear is the input from the karaoke machine mic.  And
that's kind of the point, eh?
Must I hook up external speakers to hear this? 
I'm thinking there should be a way to pipe it through my TV.

Well, you're probably using a VGA connector, right? the red, white,
and yellow cables. In the back of your DVD/VCR, there should be the
ports for all of these. plug Yellow into Video, and white into L
Audio, and red into R Audio, or if there is only one audio port, plug
the red into it. Now, do the same thing, connecting your VCR/DVD to
your TV.

If this doesn't solve it, then give the exact details of you connectors.

Good luck,
Sparky

Test your microphone by connecting it to an old cassette recorder or
to your computer's audio input.

I would guess that your microphone is non-functional.  Some mics use a
tiny battery which could be dead.  Your mic, however, probably doesn't
use a battery.  Some mics have an on/off switch - check that.

If the mic works, then make sure it is properly connected to the audio
input of your karaoke machine.