The PSU is actually fine, although for a budget this size he should do better. It's built by ATNG and the general consensus by those in the know is that it's OK. The 700W version may be pushing the platform too hard and I would not recommend it.
Here's a thread where we briefly discussed the PSU and also a link to a review http://www.jonnyguru.com/forums/showthread.php?t=6780
Might also want to switch the HDD to Samsung Spinpoint F3 1TB - about the same price but faster.
And agreed with the others above that for those games at fullHD resolution you want quite a bit more juice than 5770.
The 6950 is a good suggestion.
As for the memory, depending on how fraps records stuff the 8GB of memory might actually be handy.
If it for example stores it in memory first. After you do get the after market heatsink lower profile heatsinks are in order. The ones on the RipjawsX's for example should be low enough.
Here's a build that's in your budget that I posted on here earlier today that you might want to look at. Add that monitor and Windows 7 to this build and your set to add another one of those gtx 560's later on for SLI.