You might get away with a HD6670/HD7750/HD7770 with the 300w PSU as long as you don't even attempt to overclock the GPU (not even a little).
However, as that is an older system, I would recommend you invest in a quality PSU as well and install at the same time as the new GPU. PSUs can and do wear out.
Something like this Corsair:http://www.newegg.com/Product/Product.aspx?Item=N82E16817139028
Last, spec for your system can be found here:http://h10025.www1.hp.com/ewfrf/wc/documentSubCategory?tmp_rule=49046&tmp_task=prodinfoCategory&cc=us&dlc=en&lc=en&product=3356598
That is a x16 PCIe slot and I wouldn't recommend shoving the end of a screwdriver into it as you could damage the contacts.