Lots of people have build "VU Meters". I've done a VU meter "lighting effect". It's not too difficult, but you would need a real SPL meter to calibrate your meter for SPL. You can probably use averaging instead of RMS (take a look at the "smoothing Example") or you can measure the peak (say the loudest sound within the last few-seconds) if you wish.
If you want a simpler-cheaper approach, there is the LM3916
, which is a VU meter driver chip that doesn't require any programming. EDIT -
Sorry, that's a string of LEDs... You'll need a 7-segment displays or an LCD display.
Real SLP meters usually have A-Weighting or C-Weighting which involves some frequency filtering and timing/averaging to simulate what we hear as "loudness", rather than just measuring a sound wave.