How to access the data on your iPhone when it is off or stuck in a reboot loop?

I don't need to get it out of the reboot loop because that is hopeless since i'm pretty sure it has a virus, but i was hoping I could at least back it up or get some data off it so when I get a new phone I don't have to start from scratch. Would putting it into something like DFU mode help with this? It is a 64 GB iPhone 5 with at&t. It is running ios 7.0.4 and it has never been jailbroken. I have managed to start it up once (incredibly) but that was a few days ago and now it won't let me. When it had started up, it constantly rebooted and the screen had a fuzzy white tint around the edges of the picture.

Yes there's I think. Https://support.apple.com/kb/HT1808 or http://osxdaily.com/2011/01/08/iphone-recovery-mode/

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