First I would still try going to an Apple Store near you to see if they have any outstanding recalls.
I suspect you either have a bad CPU or the charging logic is damage here. Is your MagSafe adapter cable worn or blistered? If it is that could have been the cause. A short in the cable can damage the charging logic and the power rails. I would at least pick up a new charger before going to apple ;-} if it's looking ratty.
But before you go you could take your SSD drive out and put it into an external case to either copy off your data as well as wipe it down using a second Mac system. Here's the needed case: [http://eshop.macsales.com/item/OWC/MAU3ENPRPCI|OWC Envoy Pro].
Even if Apple doesn't have any outstanding recalls. The Apple bench rate will be the cheapest repair.