I actually have the same problem. My 5S is about 9 months old. Just yesterday it stopped charging. I've tried 3 different adapters, outlets, restarted my phone a few times. Then I plugged it into my computer, it backed-up my phone just fine but it refused to charge. I then brought the phone into an Apple Store. I originally bought my phone in London, but I've recently relocated to Shanghai, and the "genius" told me that my phone would NOT be warranted in China because "it's out of the region". If I need to get any help at all I would have to go home. The Apple call center was actually more helpful. They said that unless the phone was bought in Hong Kong, Macau, or China, it will not be warranted in the mainland China region. So the closest place for me to get my iPhone 5S serviced would be in Hong Kong, which is a 2 hours flight from where I am right now.
Is there ANY other way that I can fix the phone myself like right now? because flying to Hong Kong or London over night would not be an option and my phone is about to run out of battery...
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