Yes, you can do this.
In order to keep your app running all the time, even when the iPhone is locked, you could build a Launch Daemon. A launch daemon can start when the phone does, and is not subject to the restrictions that iOS puts on sandboxed apps, installed to /var/mobile/Applications/
.
You do need to have a jailbroken device to take advantage of Launch Daemons. Here is a good tutorial on building one.
Launch Daemons are also a normal part of OS X, so if you need more information, you might try consulting the OS X docs online. Most aspects of Launch Daemons work the same way on a jailbroken iPhone.
You'll also want to be able to detect certain events, to activate your app. You certainly don't want to be processing an audio stream at all times (maybe you only activate the app when you start walking with your cane). To detect events, like a home button press (or however you want to activate your code), I would take a look at RPetrich's libactivator library.