Iphone 5 won't charge. I need to turn it on to back it up

My iphone 5 will no longer charge though in the past few days I've messed with it a bit and gotten it to charge briefly but it doesn't last. I live in a place with no Wi-Fi and my phone hasn't backed up in a very long time; I want to get my phone to turn on just for a minute to back it up this time. I have a replacement Iphone already but haven't activated it. Any ideas/tricks on how to get my phone turned on quick?

All I can think is that it could be the cable that has a break in the wire inside. Try changing the cable for a new one, and see if it will charge then. If this fails, take it to the shop you got it from and ask for help there. If there's an Apple Store near to you, they could also assist you.