Ok sorry. Thinking about my post again I realize as usual you ask me what time it is and I tell you how the clock works. Just don't ask me if it's going to rain because I am an amateur severe weather meteorologist and I will tell you about the weather -
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Here's the short answer for the quickest and easiest way to record from a karaoke machine to a computer. Get a simple cable from Radio Shack that has the RCA jacks on one side and a stereo pc type jack on the other. Plug the black and red RCA into the left and right output rca jacks on your karaoke machine, and plug the video (yellow) to a tv or monitor. Note if you have a Tv tuner type card in your pc that accepts tv input you can plug the yellow cable into that. Plug the other end of that cable (single plug) into your computer sound card line in jack. Note you may have to adjust levels in volume control.
Download Audacity (search Google for it) and install - it's free.
Start Audacity, monitor, turn on karoake machine, with mic plugged into it. Start a track (record) in Audacity, hit play on the karaoke machine, and start singing.
This is the basic approach, and as you see (previous post) there are lots of options to try and improve sound or control over sound.
Note that using this approach you are probably listening to the sound via your pc speakers, or directly from your karaoke machine. If you have a headset (head phones) and you can plug that into your pc or karaoke machine and sing by listening that way you will probably get better sound and with less feedback, noise, and hum.
EDIT: Somehow I missed your update post about what you have. But now that I see that I'm wondering when you say you have a cd if you mean one that already has the music with the vocals recorded on it, or a cd or cdg with backing musical tracks. If cdg - then what I already told you works. If cd with voice, then as others mentioned you need to put the cd in your computer and copy the files to your hard drive. If they are not .mp3 format they need to be converted. I think Audacity will do that too - you may have to download a plug in. Once in MP3 format and small enough just go to Singers Showcase and upload. Without paying you are limited to a certain size ( I think 2Mb) and up to 5 songs. There are free compression programs to make larger songs fit under the 2Mb limitation.