IPhone 6 charge connector not initiating charge

See if there is any lint collected in there -

Also try to gently blow into the slot and clear out particulate matter that may be accumulated from every day use

If no help then consider

Make a "Genius" appointment at an Apple Store, or go to another authorized service provider.

If the plug does not fully insert, there is obviously something blocking its entry. Lint is not immediately apparent, because it will be rammed flat to the bottom of the port. Try probing with a wooden toothpick or similar.