Thanks for helping me with this. I did plug the P12 connector that you can see from the XPS 420 manual. I'm I using the good one ? Do I really need to use the 4-pin molex instead ?
The P12 is the one you want, that is specifically for the GPU.
Could very well be an insufficient PSU. The 5770SO calls for a minimum 450W in the manual. Although some 375W units may work, yours included with the PC may be inadequate.