Here is a possible approach (high level description):
You could have a simple micro controller (MCU) in a basic circuit with a number of LEDs connected. LED connection circuitry should be easy to find. Most MCU manufacturers will also give a sample test/LED flash circuit in the product datasheet, along with a sample program.
The MCU could interface with your PC via a parallel port (I have used an ATmega8 for which the input pins were level compatible with the parallel port pins). It could also interface using a more sophisticated/higher level approach - via serial/usb port using a UART to serial or UART to USB module with your MCU. You could then output different values via the parallel/serial/usb from a PC application to the MCU, and have the MCU code flash the LEDs as required. There are some articles around that power LEDs from the parallel port itself though this is not recommended. I have used inpout32.dll for accessing the parallel port registers directly. The .NET framework provides a SerialPort class you could use if not using the parallel port approach.
As for the music side of things, I've never dealt with audio files in software. There should be some libraries around and you could perhaps scale amplitude/frequency variations over time to a numerical scale displayable on the LEDs you have. Since this implies you want a basic audio player I am sure you could find some info on the audio side of things for PC apps.
EDIT:
With my answer I assumed this is for a personal project and proposed an approach that would be easy on the budget - a few $ at the local electronics store should get you everything. This is a good start, but as the other answers suggest there are likely better solutions that scale. Starting with a kit - pre-made MCU board might be easier but more expensive! It would likely be worth the cost if it is for a professional project though!
Good luck!