GraphicCardProblems :
clutchc :
Which model 760 is it? http://en.community.dell.com/cfs-file/__key/communityserver-discussions-components-files/3514/7245.760.jpg
The biggest one, minitower.
If this is the same nameplate as on your PSU, you only have a single small 18amp +12V rail for powering the entire PC.
http://g04.a.alicdn.com/kf/HTB1y5jEIFXXXXbkXVXXq6xXFXXXk/For-Dell-Optiplex-760-780-960-980-Desktop-DT-PSU-255W-Power-Supply-AC255AD-00-C112T.jpg
The HD 6570 is a 50W card maxed out. Depending on the wattage of the CPU you have, the HD 6570 may be close to overloading the PSU. If you notice the PC shutting down or rebooting, that may be a sign that the PSU can't keep up.
I'd probably chance it if it were mine, but you can make your own decision.
As an alternative, the newer GT 730 64 bit DDR5 is only a 25W card and is slightly more powerful than the older HD 6570.
http://gpu.userbenchmark.com/Compare/AMD-Radeon-HD-6570-vs-Nvidia-GeForce-GT-730/m8207vsm12582
Example: http://www.newegg.com/Product/Product.aspx?Item=N82E16814133542