Here are the troubleshooting steps recommended by Apple: https://support.apple.com/en-au/HT204267
These are the steps I personally would take to diagnose this issue and resolve it:
1: attempt to reset the SMC: https://support.apple.com/en-au/HT201295
2: try a different charger, also attempt the first step again
3: this step and all that follow require tools and technical knowledge and understanding. If you don't have both, don't try it. The damage you can cause your machine or yourself isn't worth the trip to an Apple Autherised Service Provider or store.
Disassemble the machine and inspect it for liquid damage, or burn components
Disconnect the battery - the method to do so will vary from model to model.
Ideally you'd want to try also replacing the battery with a known good working battery but it's unlike you have a spare.
4: Reseat the AC MagSafe / I/O board connect/s to the logic board. Also reseat the screws for the board to ensure its not shorting out. Again it would be worth swapping this part for a known good working part but again probably not something everyone has on hand
5: remove the logic board, I/O or MagSafe board and the fan/s. Find the power on pads and bridge them to "hot wire" start them (with the charger connected). This removes all the other possible components that could be causing an issue. If the fan spins up it means the machine is shorting out with one of the other components - enjoy finding out which one it is.
6: if you've made it to here you probably have a faulty logic board. With Apple gear its generally cheaper to just buy a new machine at this point