Sometimes when I am playing Skyrim, my computer crashes, the monitor says it finds no signal, and the speakers make a weird noise. When I reboot the computer, the bios says the CPU temperature is 60 degrees Celsius, which I don't think should cause it to crash. I have done no overclocking and don't know what is causing the computer to crash. Any input would be appreciated.
Sounds like it is running hot and shutting down. I would get a new cooler like a Corsair H Series. For now I would clean and apply a new thermal compound like IC7. Make sure your drivers are up to date.
That's not a lot of information to go by, but i would look at your temps while playing Skyrim, even if your BIOS is reporting 60c then i think it would still be best to check in game as your BIOS is giving you a reading without much/any load so during gameplay i recommend seeing how high it gets before shutting off. Also check GPU temps as well.
Also try other resource intensive games if you have any, BF3, Crysis etc and see if the same thing is happening with those games, as if it is only just happening in Skyrim then it's a different issue altogether and i'd probably recommend installing new drivers in that case as my first troubleshooting effort.
How would one check temperatures from anywhere besides the bios? This is my first build, so I wouldn't know if its software I need to download or something that sould already be on the computer to check the temperatures.
Also, how much can airflow in the case affect the CPU temperatures?
Not much, most CPU heatsinks have there own fan/s that move enough air to the point where adding case fans will only at most generally result in a 1c-2c drop in CPU temperature if they're placed properly, case fans make little difference in CPU temps, they're really only a factor for components that need to be passively cooled.