A good temperature monitoring program like Speedfan (http://almico.com/speedfan452.exe) will give you several temperatures in addition to the hard disk. I'm not sure at what temperature a system will shut down, but 60 is certainly on the warm side.
I think you need to disassemble it in order to clean the fan properly. There are several guides online but unfortunately not here on iFixit. (Perhaps you could write one when you've finished!) You have to take the motherboard out to get access. Don't take the heatsink off the processor unless you have to, as you will need to reapply new heatsink paste if you do, and you may make it worse if you don't do that properly.
With the motherboard out you should be able to separate the fan in its housing from the cooling vanes - usually there are several small screws and a piece of tape to seal the gap between the fan housing and the cooling vanes. If you find a lump of matted dust and fluff on the cooling vanes, that'll be your problem. Otherwise, or if it's no better, it may be worth renewing the heatsink paste. Get some of good quality and watch a few videos first to make sure you know how to do it.
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