If you live in the U.S. or China, Apple started the replacement process last week. Yesterday, Apple expanded it to other countries including Canada. If you qualify for a free replacement, you can return the phone to an Apple Authorized Service Provider, make an appointment with an Apple Store to bring it in, or contact Apple Technical Support online.
To get your iPhone 5 ready for the battery replacement, you need to back up your data to iTunes or iCloud, disable Find My iPhone, and erase data and settings by going to Settings > General > Reset > Erase all Content and Settings. Lastly, if your device has cracked glass or any other damage that affects battery replacement, these problems need to be repaired before Apple will accept the phone and replace the battery. In some cases, Apple says it might have to charge for the repair.