You didn't tell us which computing platform you're on.
On the Mac, I'd do it this way:
1. Use a browser with a YouTube downloading extension (such as Opera) to download the video, probably in mp4 format for best quality.
2. Use Audacity to import the audio from the mp4 (assuming that it can "pull out" audio from an mp4 file).
2a. If Audacity can't do that (I don't use it), I'd use a conversion utility to "strip out" the audio from the mp4 file, probably in .WAV format. Audacity can certainly import that...
That won’t work until you separate the audio.
On a Mac, use QuickTime to export the audio only from an mp4. QT 7.6,6 is easier to use for that. It’s free and downloadable from Apple. It installs in the Utilities folder so as to not conflict with QT10, required by the OS.
1) Screenshot of Audacity 2.1.1 settings I use for recording YouTube vids to Audacity on my HP2000 laptop running Windows 8.1
I download a YouTube vid (.mp3), than play it while running Audacity. This recording method needs a lot of volume from the speakers (more than is healthy for my ears if wearing headphones, so I disconnect headphones). I will do a test recording, and increase the computer speaker volume while running the test - the goal is to get max Audacity volume w/o getting cutoffs due to high volume transients.
It's also possible to record using the computer's built in microphone (which listens to the speaker sounds coming out). This method gives more fuzzy noise than the above method, and of course also picks up room noise (e.g. me playing along with the vid.....wonder who made that clunker....:-).
2) Screenshot of Audacity 2.2.2 settings I use for recording YouTube vids (.mp4) on my older Dell with Windows10. In this case, the recording level is set by the computer, apparently, since it does not respond to my changes to computer volume. The recording is lower volume than the Win8.1 procedure, but still usable for my purposes (slowing down song or changing the pitch).
Well, rats. the uploaded screenshots are more blurry than I expected. The relevant Audacity settings are:
I just tried what BrooksMT suggested for Windows 10 - it worked for me! After doing the recording do "Ctrl a" to select the whole recording. Then go to "Effects - Normalize" and select "OK" in the dialog box - Audacity will automatically increase the volume of the recording. Then you can export the recording to an mp3.
Audacity has a ton of stuff it can do, I have not explored much of it. I have used the Effects/Noise Reduction, particularly to remove the buzz when I record myself using the computer's built-in mike. I notice that the "vitality" of a recording is reduced when I reduce noise. Probably should play around with the noise reduction settings.
Just a reminder - save your original recording or utube conversion before working on it. For max safety, I then save a copy of the original Audacity (with Windows)...this has saved me from goofs while editing with Audacity :-/ I usually save my edits with Export Audio, but also use Save Project As