Sounds like you have a charging logic failure. Lets work this out from stem to stern. First lets make sure the MagSafe is the correct one and also in good shape. Review this Apple T/N [https://support.apple.com/en-us/HT203207|Apple Portables: Troubleshooting MagSafe adapters] Also inspect the contact pins and the mating surfaces on your system if dirty use a cotton swab (Q--Tip) slightly dampened with 85% isopropyl alcohol to wipe them down.
Now lets plug it in and tell me what the color of the MagSafe cords LED bright Green, Amber, of dim Green.