I would like to record from the device microphone and then be able to calculate the beats per second of the recorded audio, is it possible and how? is there any open source project I can look?
To record music from the microphone, use the MediaRecorder and store the sample (a few seconds should be enough).
Finding the beats is trickier, there is a nice explanation at GameDev, but if you have no background in signal processing it will not be easy.
You might also have to use native libraries to get nice and fast implementations of FFTs (search here)
See also this related question : c++ FFT Beat detection library?