Here is a fairly short video that shows how to test battery and charging port:
If battery is completely dead after sitting awhile and charging port is not charging battery...that would explain it.
But if charging port is not charging the battery, then it could just need charging port cleaned out...it collects dust and debris like "Belly Button fuzz" and it cannot charge because the contacts are obstructed. If port is clean and still not charging you can replace the port:
iPhone 6 Plus Lightning Connector and Headphone Jack Cable Replacement
But you do not want to mess with removing the screen with battery connected or charger connected or you can mess up the backlight circuit.
After testing these, get back with an update as I check in a couple times daily as well as other good techs to give other options or points of views I may not think of :)