You know... most recording/audio editor programs (Audacity, GoldWave, etc.) have a meter display on the computer screen (with no additional hardware).
If you are going to use the Arduino and buld some circuitry, I'd recommend NOT using a computer. But, if you want to use the computer as an amplified microphone, you can simply turn-on the mic in the Windows 'Playback' Mixer and turn it up volume. Then, the amplified line-level signal should come out of the headphone jack... You can run the headphone-output into the Arduino (with the proper signal conditioning to block negative voltage swing, which can kill the Arduino).
I normally use a Peak Detector Circuit
to condition the audio input. But, you can also use a pair of resistors and a capacitor to simply bias the signal at 2.5V.Here
is a little pre-assembled PC board with a microphone and an amplfier. You can connect it directly to the Arduino and you won't need a computer. (of course, you will
need a computer during the design & construction process to to develop your code and to program the Arduino.)
Before you start making a VU meter, this simple Analog Input Example
blinks an LED depending on the analog signal. You might want to experiment with replacin the pot with an audio input.
Here is a VU Meter Example Projcet