You can't get to the motherboard to connect a cable to it (if there were such a thing, which there pretty much isn't) without opening the phone.
You have a few choices:
Connect the phone to your computer. You can copy all the files you can see.
If USB debugging is enabled, you can use adb to pull any file from the phone's storage to your computer.
You can find a friendly computer repair shop that has a pretty well destroyed Note 3 (maybe water damage to the motherboard) and they'll plug the screen into your phone in place of your screen long enough for you to do a backup. Or a shop that will replace the screen without doing a factory reset first. (Most shops do that first so that, among other things, there's no chance they can steal your data, like passwords, etc.)
If the phone is in such bad shape that it doesn't pay to repair it (even scratched and dented they're still worth decent money), the motherboard probably doesn't have any data on it to recover.