Using the iOS app you can make a video and keep the music going. Check it out:
JamCam not only keeps the music playing during video recording, but records the raw music file playing from your iPhone music app, in sync with the video! There’s no other app like it. JamCam is about capturing and sharing the musical moments in your life – all in one step.
You open the app and record the video within the app. You can set the song and it will begin playing when you begin recording and it will stop when you stop recording. It only works with DRM-free songs. You can save these videos to your phone’s camera, as well as share them within the JamCam community or within your own networks. So if all of your friends are in Instagram, you can create these videos with your music and then just share them there. (Via TechCrunch).