[Verizon Galaxy Note 3 Q&A, Help & Troubleshooting] bftb0:
Try the following and report your results.

1) Pull the battery with nothing plugged in to the USB port. Put the new battery into the phone. While watching the screen carefully, plug the ORIGINAL charger and ORIGINAL cable in to the phone.

Do you see:

(A) Nothing at all

(B) A battery charging animation, or

(C) A static battery icon graphic followed by a brief appearance of the Knox message "Knox Warranty: kernel"

Hopefully the answer is (A) or (B). If not you might have a chicken-vs-egg problem getting your battery to charge.

Answer these questions:

2) Is the original charger the OEM (Samsung) 2A Wall-wart charger, or something else?

3) Perchance was the new charger identified as "Apple Compatible" or "For iPhone", or similar?

4) Are you able to use both cables (old & new) for data connections to a PC (testing with some other devices that can exchange data)

5) What are the rated charging capacities of the old & new chargers? 2 Amps?

The business about Apple compatible chargers is that Apple intentionally violated parts of the USB specification (to suit it's own needs) that detail how devices are supposed to interpret signals on the 4 wires of a USB 2.0 connector during initialization. So a lot of "Apple compatible" chargers produce out-of-spec signals during initialization, and many Android devices will conclude that those chargers can only provide 500 mA of current. So even if they are rated at 2A, the charge controller in the device won't pull anything more than 500 mA.

Similarly, there's all sorts of counterfeit junk being sold, even USB cables. I bought a "USB 3.0" cable in a box that appeared to be Samsung's from Fry's electronics. Despite it having blue connector inserts, it didn't even have any D+ or D- connections. It was neither USB 3.0 nor even a data cable - it couldn't even be used for charging beyond 500mA (because of the missing D+/D- connections). If you can use the cable for communication, at least you know that all four wires are present.

The amount of current that gets pushed into a handset battery during charging is typically controlled by a charge controller chip that sits either on the motherboard or on the (replaceable) USB connector daughterboard and the battery, and also watches transitions taking place on the D+/D- lines shortly after plug-in. (The external charger is just a dumb power supply rated to provide 5v up to whatever it's maximum current draw is before the voltage collapses... unless it is an Apple charger, and then it wiggles the D+/D- lines around inappropriately, confusing any device you plug them into that actually follows the USB spec, aka all Android devices)

Were it not for some certain odd behaviors that the Samsung bootloader engages in when there is an unsigned boot image flashed to the device, the scenario I'd be most likely to suspect is the following:

That you have a bad charge controller chip and you need to replace the USB connector interface. (IIRC, I think the charge controller chip might be on the connector module behind a flex connector, but I can't remember).

Sorry for all the questions - just trying to eliminate possibilities which could be interfering with your debugging.

https://www.ifixit.com/Teardown/Samsung+Galaxy+Note+3+Teardown/19209