If you have a super clean sine wave signal with no distortion, you might be able to do this with a pic that is also performing other tasks. Otherwise, it is a very difficult task involving FFT or DFT programming--certainly not the standard place to start learning PIC programming. (And PBP is a pretty code-heavy compiler, so it may not be able to process the data fast enough, even as slow as 150Hz.)

The biggest drawbacks I see to the NE567 is high voltage requirement and high current draw. On the other hand, it is the perfect simple solution to a potentially challenging problem.