Hopefully something from your old system is salvageable.
You also didn't need that extra wattage of the HX620 in your current system. It will run stable on less.
As for a new power supply, with your proposed build a good quality PSU at 500W will be adequate. The single component that can draw the most power is the graphics card, so bare in mind if you decide to change your mind from a 570 and get say a 580 you would need to increase the PSU to about 550W. Likewise, if you go down the SLI route, 2 570's will command a good 600-650W for safety.
As for brands and recommendations, I have always used Seasonic for the last few PC's. Not the cheapest, but very good quality power supplies. Anything from Corsair's TX or HX series is also a good choice, and come a close 2nd to Seasonic's PSU's. You generally wont get a bad PSU from Antec, Be Quiet!, Cooler Master, Enermax (expensive) and Zalman neither - Your not short on choice. I would probably choose one that's modular too, and make sure it has at least 2 6-pin PCI-E power cables (Your proposed GTX 570 needs a pair of these). Most mid-to-top end supply's will come with 6+2-pin PCI-E power cables, and are better still as they can power top end graphics cards like the GTX 580 / Radeon 6970 should you decide on one later on.
Personal recommendation: Seasonic M12II-520 (From experience)
The Piledriver Beast - AMD FX-8350 | Gigabyte 990FXA-UD5 | 16GB Elite Group DDR3 1600 | Tri-Crossfire R9 280X | Crucial 512GB M550 SSD