How to jailbreak 6.1.3 on iphone 5?

Or at least Downgrade it

As of now the ONLY legit iOS 6 jailbreak is Evasi0n, and Apple patched the exploits they used in iOS 6.1.3, the hackers that made the iOS 6 jailbreak have stated that they will not make another jailbreak until iOS 7 drops sometime this summer - fall. So unfortunately there's no way to jailbreak ANY device running iOS 6.1.3. Your just gonna need to wait till iOS 7 gets released sometime later this year. Sorry.