The symptoms fit a few things;
1) Overheating, if the laptop does not allow airflow through the vents (due to dust buildup or other blockage) it can cause that. When was the last time the laptop was physically cleaned? If it has not been disassembled and cleaned, I'll recommend doing that, this video may help
https://www.youtube.com/watch?v=U6SHCbqeGs0 He takes it to replacing thermal compound (you might look into that) but you should really only need to get to 3:38 where he removes the fan - you'll be able to access and clean the vents/fan then. If the heatsink itself is removed, thermal compound will need replacement
2) Bad battery, Dell uses a battery system that reports the battery to the machine - if the battery is bad, it may not work. Have you attempted with the battery out?
3) Power Jack, a common problem on many laptops is the power jack where the adapter gets plugged into gets worn from being plugged in and unplugged, it can cause the laptop to not function properly. To check that, try moving the jack while the adapter is plugged in - if the light comes on or flickers, that's likely the cause.
4) Last thing to check is the charging circuitry, you'd only be able to perform a visual check for burnt areas or swollen capacitors - disassembly required also
While I'm sure I've left out a few possibilities, that would be the primary things I'd check