I'm assuming this was with an apple product, because any other laptop I've seen you won't be able to charge with cellphone charger...
Anyways, power malfunction is obvious, the extent of the damage will need to be determined by taking the laptop apart and checking each component separately. If you're lucky only the motherboard will need to be replaced, if you're unlucky, well I don't know.
I'd recommend you call up apple and figure out what they can do for you from there or take it to a computer repair shop and see what they can do for you.
I doubt you would be able to follow any DIY solution for this case.