Okay, posting system specs...
Processor 3.0GHz 2nd Generation Intel Core i5-2320
Memory Type DDR3
Maximum Memory Supported Not Avail.
Hard Drive 2.0TB SATA
Hard Drive Speed 5400RPM
CD-DVD Drive 24X DVD-RW
Graphics: 8800 GTS
Dimensions 16.12”(H) x 6.8"(W) x 16.5"(D)
Also, I do not have a PSU at all...
Are you sure? If you dont have a psu, then your system is not running at all. PSU = Power Supply. If you dont have a psu i suggest a good quality 500w unit, then you can chose whatever mid-high end graphics card you like. but yes, the 5870 will be a good upgrade to what you have, $100 is not bad, so long as the card is working properly and the fan bearing isnt making noises or anything. They are getting a bit old though and it might not last all that long.