Radeon HD 5670 on a 250W power supply

It probably can handle the 5670, but it'd be cutting it a bit too close, most decent quality 300w psu can handle a 5670 with ease, so that 250w likely could, but then if you plan on adding more drives or peripherals later it might fall short.

This will suffice: http://www.newegg.com/Product/Product.aspx?Item=N82E168... and you could borrow it to your next build when you upgrade

Yeah, it might work but it isn't wise. An HD5570 would be a better choice but just replacing the PSU is the best idea. Then you can get whatever card you might want. What is your monitor's native resolution and how much are you willing to spend total

That's really not a good PSU. Cooler Master has some decent PSUs but that one specifically is quite poor and lacks even active PFC which is the first thing you should look for in a current PSU. This Antec unit is $10 cheaper($25 cheaper after the rebate) and it is much higher quality with active PFC, 80 plus certification and 2 extra amps on the +12v rail.
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

I'm not looking to make a complete gaming rig, I just want to get rid of the crappy integrated graphics card that came with the computer, so the max I would be willing to spend would be around $100, that being said, as of now, on amazon the 5670 is only $10 dollars more, and from review that I have read it seems much better than the 5570. So should I just get the 5670 and see if my computer blue screens, or be safe and get the 5570

Well, it really would be best to replace the PSU. Even if the HD5670 does work for now it would mean you are pushing your current PSU to it's specified limits or beyond which makes the unit operate at lower efficiency and will be much more likely to fail before long.
The Antec unit I linked is really quite cheap and a good deal at $35 before rebate and just $20 after. At that price you can still get the HD5670 and stay around or under $100. With this card and the PSU you would spend $105 before rebate, $90 after;
http://www.newegg.com/Product/Product.aspx?Item=N82E168...
Or if you are ok with rebates then this GTS 450 plus the PSU also comes out to under $100 after rebate;
http://www.newegg.com/Product/Product.aspx?Item=N82E168...
The GTS 450 is roughly 50% faster than the HD5670 on average. Definitely the better of the 2 choices if you are using a high resolution monitor.

I've personally ran a HD5670 on a crappy 240watt PSU and it worked fine, It's not a clever idea for the long term though, as putting that much pressure on a PSU can shorten the life span