Looks like I'm actually useful for once!
First, your local PC guy is a schmuck. Those $500 video cards are made for rendering the human genome, not filing your taxes. What he's trying to sell you is the equivalent of paying extra for a car dealer to re-chrome your wheels: Shameless upselling.
Anywho, BSODs have a few major causes - software, heat, and power. If you're lucky (and I doubt it in this case), you might just have an issue with your Windows installation - if your computer powers up to the "POST" screen (which on most commercially made machines shows the logo), you might just need to reinstall Windows.
The largest cause of BSODs is, one way or another, heat. Heat causes problems in two ways: If a chip overheats, it won't work properly, and if it's running too hot, it will be slowly damaged over time. If your computer had failed after a few hours of use, I'd suggest that you open it up and blow out the dust bunnies, but a blue screen after being left off for several hours suggests that if heat is the culprit, the damage is permanent.
The final cause is, as mentioned earlier, power. If your power supply were on the fritz, you'd likely not see anything at all or have a crash sometime after boot - if a computer doesn't receive the voltages required to start up, nothing happens. However, in addition to the power supply, there's a whole bunch of power filtering electronics on every part of your computer - motherboard, video card, etc. - and a failure in any of these could cause a crash on startup. It is worth noting that power supplies are very cheap - odds are you can pick up a secondhand ANTEC or Corsair power supply of 400 watts or more for under $30 on eBay.
One thing to check for is whether your computer is booting, but you just can't see it. If you get a blue screen but the windows startup noise still comes through your speakers, it might just be a bad video card - a relatively inexpensive and easy fix.
If you're looking to purchase a replacement computer, the biggest question is what you want to use it for. If you're not doing anything too processor-intensive, your best bet is to buy something very cheap - you can get a fairly good desktop sans monitor for about $250 nowadays - and upgrade it yourself. So long as the computer has a PCI Express 16x and old-school PCI slot (most of them do, though you should check before buying), you can easily add quad monitor support for about $120 more by adding additional video cards like these:
http://www.newegg.com/Product/Product.aspx?Item=N82E16814139038
http://www.newegg.com/Product/Product.aspx?Item=N82E16814139018
An even better solution would be to use an "SLI" or "Crossfire" motherboard that would allow you to use two standard video cards instead of having to buy a more expensive card for the much older, much slower PCI bus. However, these motherboards are extremely rare on affordable computers, and unless you're willing to build your own system, you won't get a very good price.