ADC for PIC is very stable!
I use 10bit ADC and max error ir one bit.
Need to use 7805 for stable 5V power, inductors and capacitors for smaller pulsation.

If You need simple Audio level meter, then use rectifier, resistor and capacitor. And Use Vref pin on PIC, to set max value. Thats all.

For More advanced level meter use OpAmp and create band filter or band filters and build up to 8 band spectrum analyser. With 16 OpAmps it is very simple to build. And for example 16F877 or 18F452 works fast. You can read all 8 channels and display. By software You can add peak detectation and build log scale.