Your first link I think was wrong.It showed a list of motherboards.
Here is a review on the Radeon HD 5770 graphics card.It might even be hard to get your hands on a HD 5770 then again you might be lucky.It looks like a good choice for you and compares favorably with the older 4870 but with new features,DX 11.
Check the benchmarks in this review they look pretty good.
http://www.guru3d.com/article/radeon-hd-5770-review-tes...
I'll tell you again that in less than a weeks time the big sale will be on black friday.
Quite possibly you might be able to get better components for much cheaper than you think.
As an example though if you had a few bucks over your budget maybe $50 over your estimate I would get these components.
Here is what I think that you should get.
First allocate about $100 for the Hard Drive (I like Western Digital's) and a DVD burner.DVD burners are about $25 to $30 so you should have $70 to $75 left over for the Hard Drive.
Any cheap case can be obtained for under $50 but even a $30 case will do.
So allocate about $100 for the case and power supply.Notice that there are cases that come with a power supply.Try to get a power supply somewhere around 600 watts and be sure the power supply is compatible with the motherboard.You might get away with using a lower power 550 watt power supply on your build with just one card.I tend to spend a little more and get beefier supplies like 850 watt ones on my builds in case I opt for higher wattage and/or crossfired graphics cards.
With a limited budget you will have to opt for a cheaper lower wattage power supply most likely unless there is a big sale on a higher wattage one.Look at the power consumption with a Radeon HD 5770
http://www.guru3d.com/article/radeon-hd-5770-review-tes...
It says you need a 500 watt power supply at the minimum with one card and 600 watts with 2 at the minimum.So if you intend to crossfire 2 of these cards you will need a 600 watt or better power supply.If you can make do with just one card I see a inexpensive power supply here.Be sure to be careful in selecting a power supply with an EPS CPU connector (power to the CPU).
This one is $40 after Mail in Rebate for a 500 watt power supply
http://www.newegg.com/Product/Product.aspx?Item=N82E168...
Here is a case with a power supply.It's just one example.This one has an EPS connector.
Myself I would prefer a better Corsair or Antec power supply but perhaps this one will do
http://www.newegg.com/Product/ComboDealDetails.aspx?Ite...
This combo deal on CPU,mobo and memory about $400 actually $378 after MIR but on black friday who knows possibly $300? which would be $100 less
One of the best AM3 motherboards out there and very feature rich with easy oc features.Also with a faster AM3 AMD CPU which is very popular.
You would be so satisfied with this motherboard you just can't go wrong here.
http://www.newegg.com/Product/ComboDealDetails.aspx?Ite...
Then allocate about $180 for your HD 5770 graphics card.
That might come to around $750 but if you had to stay below budget you could perhaps opt for this CPU/mobo combo deal.Notice though that you would have to spend around $70 for DDR3 RAM
http://www.newegg.com/Product/ComboDealDetails.aspx?Ite...
That Phenom II X4 925 that you showed earlier looked good too.
Here is a combo deal for it.
http://www.newegg.com/Product/ComboDealDetails.aspx?Ite...
As advise stick with the AM3 Phenom II's rather than that AM2+ Phenom II X4 940.
Those with older AM2+ boards would be happy with the 940 as a CPU upgrade but on a new build go for the newer AM3 CPU's instead.
I'd much rather opt for the 945 or the 955 instead.Plus you are better off with a newer AM3 motherboard as well so why not get one of the best ones.