Why would a computer automatically restart itself every 5 to 10 minutes when in a PC game

There are many different problems that can cause a system to spontaneously restart. It could be that either your system doesn't have a graphics card powerful enough to run whichever game you're trying to play, although not very likely if you can play the game for a bit. It may also be that your system is overheating, you could check your fans and power supply and ensure that everything is running properly. Also check the support website for the game in question and see if anything similar has been posted by other users or perhaps there's a patch available to fix the problem. Many possible answers:
1. Overheating of the parts (unlikely)
2. Lack of RAM. Check the game case, and compare the RAM required to the system spec. (Likely - this problem can usually be solved for less than $100)
3. The game is buggy. (very likely - refer to game manufacturers website, download patches etc.)
4. Graphics card not compliant with game (unlikely unless your graphics card is more than 4 yrs old)
5. Graphics card does not have enough memory (likely with old graphics cards. Modern games now usually demand 128mb graphics cards. This may cost upwards of $140 new)
6.could be out of date graphics drivers


Minor addition:

New video cards can have a large power consumption compared to older cards and it would be wise to make sure that the power supply of the unit is at least at the wattage recommended for the video card.

Usually there are programs that can be found at the mother boards manufacturing website and at the video card manufacturing website that will allow a person to monitor the temperature or the CPU and Video card and such.