But the fan getting really loud isnt anything new, I never thought anything of it before.So I wonder how long the CPU will last for even if I do clean out the computer and replace the thermal grease.
If the CPU stops working one day what would happen? How will it affect the computer? What does the CPU even do?
82c is too hot for that CPU. As a method of self preservation the CPU will slow itself down (throttle) to attempt to lower its temperature, and might shut down altogether. Who knows how long it will last but it's unlikely permanent damage has been caused unless it's been really badly overheated. My Core i5 was briefly running at 95 degrees two years ago and I noticed it was throttling. It was dust causing it, after cleaning up the heatsink it's still going strong now with daily use...
As for what does the CPU do - pretty much everything. Without it you effectively have no computer...
There should be plenty of good video guides to cleaning heatsinks and replacing thermal paste on youtube. As for thermal paste, I'd suggest Arctic Cooling MX-2 as it's pretty cheap, works well and is very easy to apply.
If you decided to go for aftermarket cooling, you don't have to spend as much as $60 but do check whatever you get will fit in your case. The tower style coolers with 120mm fan won't fit in smaller cases for example.
Here's a handful of cheaper ones that I think work OK (and will certainly outperform a stock Intel cooler)
Coolermaster 212 Evo (big, for larger cases)
Coolermaster Hyper TX3 (medium size for most tower cases)
Zalman CNPS10X Optima (big for larger cases)
Zalman CNPS8900 Quiet (low profile for small cases)