My budget is $750 (without shipping, tax, etc.) Since I already have keyboard, mouse and monitor I don't need those. This build will be mainly used for gaming and surfing the web I'm planning to play Battlefield 3 with it. I'm running a resolution of 1440x900 so that GTX 560 Ti is probably overkill right now. But I'm going to upgrade my monitor anyways.
I also will not be using SLI/crossfire-x.
Any suggestions? Will my PSU handle it? Should I upgrade my MOBO to something better?
Get rid of that FX CPU. Any FX CPU will bottleneck a growing number of game titles and the quad core variants aren't much better than a quad core Athlon II chip. I would put the quad core FX chips between the quad core Athlons and the quad core Phenoms, much closer to the Athlons. Since gaming tends to utilize highly parallel CPUs like the six and eight core FX CPUs extra cores pretty poorly more cores does not help much if at all, depending on the game being played. Phenom II quad core chips represent the best of AMD's gaming performance right now. They are pretty far behind Intel since they are three or four years old so yeah... not good for AMD.
Battlefield 3 is very CPU intensive and will be bottlenecked by pretty much any AMD CPU. The quad core Phenoms may be okay with it if you overclock one of them (Phenom II x4 955 BE or better) but otherwise you would be much better off with a recent Intel quad core chip, preferably the i5-2500K for overclocking and the i5-2400 for non-overclocking systems.
If you are upgrading your monitor to something capable of higher resolutions than 1920x1080 then you will be better off with a similarly (or better) performing video card with more than 1GB of video memory.
Due to your constrained budget I would not recommend this but I will recommend switching to a Radeon 6870. It has similar performance to the GTX 560 ti but can be found much cheaper than the GTX 560 ti.
Grab a Radeon 6870 for about $170 or less. You can also get the Cooler Master Hyper 212 CPU cooler instead of your choice for about $20. It will still take decent overclocks and can get the i5-2500K up to around 4.4GHz easily enough and maybe a little farther with a second fan installed for a push-pull configuration. I own this cooler and put a second fan I had on it and it works great for me. If you get the i5-2400 instead of the i5-2500K then you can either get the hyper 212 or just stick with the stock cooler that comes with the CPU.