You have a few hurdles to overcome -
firstly the lcd display you have chosen sounds like it doesn't have an intelligent driver - the HD61202 needs a clocked stream continuously. I suggest you choose a display that has an intelligent driver such as the toshiba T69??. That will make interfacing much easier. I believe Mr Karas has some sample code for driving such displays.

next is the issue of the analog input and sampling this at a sufficient rate. Along with this is the conversion from sampled time to frequency domain. There are many techniques for doing this. All are non-trivial.

If you want the bouncing bar graph display like on ghetto blasters and suchlike, they use ics specially developed for that purpose. They probably use a combination of analog filters and digital processing to keep the cost down.