My power supply: CHIEFTEC CTG-750C
...
675watts available on 12v rails...it SHOULD be able to hold a 5700 quite comfortably.
But the question shouldn't be can it power your new GPU, but 'can you feel safe using a low-cost PSU of uncertain build quality'. It doesn't appear to me be tagged with any 80+ certifications...that "85+" badge I do see looks a bit sketch to me. I have to think most of the PSU guru's won't like it and will recommend getting something better.
Also, if more than 3 or 4 years old components inside have probably shifted values enough that it's not even performing up to the spec's it originally met.