Hi Raptorius (everyone),
Before you go any further, I'd suggest you follow what sr103nx has recommended (see if you can test the 7870 in a different computer). This is a quick way to narrow down the issue. By the way, a 600w PSU is fine.
It seems to me that your computer was using an integrated video card (not an discrete video card - a.k.a. dGPU - one that plugs/slides into a PCIe Slot). Did you have to pull anything out of a PCIe slot before you installed the new 7870? If not, then we can say that you do have a video card integrated on your motherboard (a.k.a. iGPU). Assuming this is correct, you will have to go into your system BIOS and disable it manually (again, sr1030nx has provided you a nice step-by-step process).
That being said, have you tried calling Dell Tech Support for help? Or do you still have the user manual? It should have information on how to disable the iGPU and install a new dGPU.
And here is a quick check list for installing a new dGPU (one that goes in the PCIe slot):
1. Make sure the 7870 is compatible with the motherboard
2. Insert the 7870 in the available PCIe slot
3. Make sure it fits nice, even and tightly
4. Plug the power cables from the 7870 to the PSU (by the way what's the name and model of the 7870? Gigabyte, MSI, XFX...)? If I recall correctly, the 7870 video card has two 6-pin power connectors, so you may need a y-splitter to connect it to your PSU)
5. Firmly connect your monitor to the 7870 (make sure the DVI cable you're using is connected firmly to both the monitor and the video card) - and of course connect the power cable of the monitor to an power outlet (doh
)
6. Turn on your system and make sure the fans on the 7870 spin (if not, there is either something wrong with the video card or you haven't properly connected it to the PSU) - leave the case open so that you can see the fans spin.
I hope the above helps you, and let us know how things go