iPhone 5 has a faulty battery, Apple said they'll replace it for free, but I can't find out how.

Actually it does.

"The iPhone 5 serial number you entered is eligible for this program. Please choose one of the service options below.

Replacement process

Choose one of the service options below to have your battery replaced. Your iPhone will be examined prior to any service to verify that it is eligible for this program and in working order.

Please call your service provider to confirm that battery replacement service is available on the day you visit them.

• Apple Retail Store - Make an appointment here.
• Apple Technical Support - Contact us."

That's exactly what is says when I enter my serial.

Again, I can't really go all the way to a retail store as there's not any near to me, and contating technical support just links back to the normal tech support page, which has nothing to do with my battery issues.