I haven't fully set up a list of power supplies yet, so my apologies for not being able to recommend something cheaper. This PSU will do a significantly better job than the one you linked. (the website is a price aggregator for vendors). At the best price, it's also cheaper than the one you linked. It's a Tier 2a 550W Bronze Certified Power Supply for $55 ($45 on promo), with 44A on the 12v rail.
http://pcpartpicker.com/part/xfx-power-supply-p1550sxxb9
When upgrading a system for gaming, the first thing you upgrade is the power supply (PSU). If you try upgrading a performance part (CPU/Graphics) before the PSU, the old PSU will explode. The primary ability of a PSU that you are looking for is not Wattage (i.e. 450W vs 550w vs 750W), it is the 12v rail Amperage. CPU/Graphics use the 12v rail.
[edit] Sorry for the edit, deleted stuff before I realized my math.
After the PSU, you'd upgrade a performance part. Upgrading the CPU would require upgrading the motherboard in most cases, and thus you'd upgrade the Graphics card, normally. The problem is your Sempron is awful, and that PC has you using DDR2 RAM, and a tiny amount of it. If you were to upgrade all of this, you'd be above your $250 budget. As a result, I'd have to recommend upgrading your CPU, and going the APU (CPU+Graphics on the same part) method (the route the PS4 and XB1 went). This requires a different motherboard.
APU -
A8-7600 - $98
Motherboard -
MSI A55M-E33 - $46
RAM -
AData 1x4GB DDR3 1600 CAS 9 - $35
PSU -
XFX TS 550W Bronze - $55 *previously mentioned*
--------
Total - $234
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