If you've tried different chargers, cables, and replaced the battery then it can only be a problem with the phone itself (more than likely the port or the hardware connected to the port).
Samsung has a 1 year warranty for their phones, so if you are within that window you should get a hold of them and see what they can do.
If not, then there are shops that can probably fix it, but it will probably not be cheap. Worse comes to worse, a used Note 4 can be had for around $370 now.
Edit: I suppose you could back up your photographs and then try a factory reset, but that only has a small chance of working.