Assuming that this isn't under warranty?
Try the above, though I doubt a dodgy HDD is going to cause a spontaneous reboot at POST (dealt with hundreds of dead/dying HDDs and never seen one do that to a machine that I can recall, but without a video I can't see if its actually getting past POST). Up side to this is that it usually means that your HDD contents should be OK (so the enclosure mentioned above is a good investment). If you remove the HDD and try and start the laptop you should just get a message about not being able to find an operating system.
RAM maybe, so worth a shot given the spontaneous reboots you've reported while using it, but normally the machine would attempt to boot and get funky rather than just fall over during POST. Still worth a shot. At least RAM and HDD are easily accessible on laptops (with some exceptions).
Overheating is a pretty common laptop issue - especially if you use it in bed, on lap, or anywhere that isn't a flat surface. What kind of fan noises have you had out of the machine - a lot or none? Can you feel airflow around the side vents? If the fan has died or is otherwise not functional, you'll get it to cook quickly.. though starting from dead cold (ie leave it off for a few hours), should allow you to progress further before it has a fit. If it can be narrowed down to overheating, that's really a tech job - unlike PCs, laptops are a lot more delicate, and far more people can dismantle them than can properly reassemble them (including half the kids from manufacturer repair agents....).
Assuming it's not just a dead fan, a dodgy RAM module or a freaky HDD, then you're rapidly going into the territory of repairs that may well hurt the budget.. a lot... as there aren't many parts in laptops that can be replaced, and beyond the hard disk and RAM modules those parts are model specific and bloody expensive. In such a case.. always ask for a quote to repair (and acknowledge what the quote fee is given its most of the repair labour to find the problem).