The problem is most likely from the PSU. Before you buy other PSU, use the PSU from dell Inspiron 570 to see you can boot the PC or not, because you said the HD7870 worked with dell Inspiron 570.
If you can boot the PC with that PSU ( from dell Inspiron 570, the dell OEM PSU is good one), you know you will need the new PSU for the HD7870. The wattage, even the 775W, does not tell you the whole story about the PSU, only the good brand is good.
Here is how to buy a good PSU. http://www.tomshardware.com/t/power-supplies/