Sorry to be the bringer of bad news, but judging from the Amps on the 12v rail i doubt your current system would manage with say even a 5670.
that 15A on your 12v rail is roughly 180watts. You should know that the ''main'' parts of your pc rely on the 12v rail, (gpu, cpu).
the 5670 alone is about 60-70 watts i believe, which leaves you with 110. Then taking into consideration that the cpu is also roughly the same in terms of wattage, means you have a total of at least 140 watt on load. That's not a number i'd feel happy with running on a psu that's rated 15amps on its 12v rail
In short, to do a gpu upgrade the current psu needs to go, i'd recommend an antec earth-watt 380, they are relatively cheap and are great for mid-tier systems.