Thanks for the info and the photo! Here are my thoughts:
You have a very powerful CPU capable of handling those GTX 1060 and RX 480 (even a GTX 1070 and GTX 1080, if you so choose). But having a 1080p/144Hz monitor, your planned GTX 1060 or RX 480 *might* struggle to reach more than 60fps in *some* games (especially GPU-hungry AAA games with all the eye-candy settings enabled). More often than not, it is recommended that for 1080p/60Hz Ultra the GTX 1060/RX 480 is enough; while for 1080p/144Hz *or* 1440p/60Hz, the GTX 1070 would be more sufficient.
In any case, choosing between the GTX 1060 or RX 480 given your current monitor and resolution would still be a logical choice, esp. if budget is a factor.
Now, my concern is that PSU which only has a 324W combined +12V rail power, i.e., equal to 27A. Not to mention, it's a no-name PSU (AcBel, actually) that can be construed to have questionable quality, reliability, and efficiency.
Your i5-6600 has a TDP of only 65W. A GTX 1060, depending on the exact model you choose, could require a 6-pin PCIe connector which can mean a possible power draw of 150W max. If the GTX 1060 would require an 8-pin PCIe connector, then the possible theoretical power draw can reach up to 225W max. Same goes for the RX 480.
So, your CPU + GPU could draw 290W of power at that +12V rail (not yet including other components such as HDDs, fans, RAMs, etc.). With other components to consider, we can say that your rig can draw around ~300W++. This means your PSU ain't gonna cut it.
I highly suggest getting a good quality PSU of ample wattage but with enough headroom (in case you decide on upgrading other components that would necessitate additional power). I'd look into the Seasonic G-550 or EVGA SuperNOVA G3 550W as best options. You can also get the more affordable Seasonic M12II-520 EVO if budget is limited.
Between the GTX 1060 3GB and the RX 480 4GB, I'd personally choose the RX 480 (with higher VRAM, for a lower price).