Very Cheap Build

Run many games at very high settings. You should be set really. Just make sure you have enough RAM.

Should 4GB be good?

4gb is good. I personally recommend 8 gb. They're not expensive, as long as you get a good deal. Tigerdirect has a sale on an 8gb patriot bundle for 55 dollars I think. Better than spending the ridiculous 120 some brands sell it for.

http://www.tigerdirect.com/applications/SearchTools/ite...

http://www.tigerdirect.com/applications/SearchTools/ite...

same price, same brand, higher mHz on the 2nd link.

I've read games only can use the max of 2 gbs of ram, so if you have 4gb it should be more than enough.

8Gb isn't much more, if not the same price as 4gb. So why settle for less? These will be good for upcoming years.

How big of a PSU will I need?

4 GBs of Ram is fine for gaming and most applications. THG did an article on it.
http://www.tomshardware.com/reviews/ram-memory-upgrade,...
that being said Ram is rather inexpensive right now.

your above build is well suited for a budget gaming build. that 6870 has a very good performance/price point

I would put this quality budget minded ram with it
2x2gb
http://www.newegg.com/Product/Product.aspx?Item=N82E168...
2x4gb
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

now all you need is a quality PSU and HDD
PSU:
http://www.newegg.com/Product/Product.aspx?Item=N82E168...
http://www.newegg.com/Product/Product.aspx?Item=N82E168...
http://www.newegg.com/Product/Product.aspx?Item=N82E168...
HDD
http://www.newegg.com/Product/Product.aspx?Item=N82E168...
http://www.newegg.com/Product/Product.aspx?Item=N82E168...
pick whatever DVD/CD drive that is rated 4 or 5 eggs and has free shipping and a case that you like and you are in business

Would this PSU: http://www.newegg.com/Product/Product.aspx?Item=N82E168...
and this HDD do it?: http://www.newegg.com/Product/Product.aspx?Item=N82E168...

if you are tight for cash, buy this one now. easy choice:
http://www.newegg.com/Special/ShellShocker.aspx?cm_sp=S...

not sure that would be a good gamer. he will still need to add a GPU, as the A8's built in graphics are better that the intel ones and the ones on most MB's they still aren't good enough for gaming. the combo comes with a single stick of memory, yet the MB and CPU support dual channel, this will slow it down also.

this is a bench mark I found for the A8-3850 against several CPU's using a GTX580 GPU, so the differences were CPU alone. The i5-2400 is not in the test but it performs just under the x6 1100T for gaming.
http://www.winmatrix.com/forums/index.php?/topic/32022-...

the PSU is over priced, this one is has better reliability, its and modular:
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

the Hard Drive is a solid one, but very slow compared to the newer drives. this one is very fast:
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

the price of your picks = ~$120
the price of the above suggestions = $120

^the link changed jerred, my bad. it was a $35 Antec NEO Eco 520w shell shocker. SOLD OUT.

I'm not sure which planet you live on, but on mine when a product as something that is more than another product it is never the same price.

ddr1333

4gb is 29.99

8gb is 51.99

That's a 22 dollar difference, it may not seem like a lot when talking about 100+ gaming cards, but that 20 bucks can go towards something more worthwhile, better case, cooling, better cpu, better video card, and actually give you a better gaming experience.

When the time comes for him to need more ram, either he can upgrade then at a cheaper price, or his parts will be old anyways so he will need to upgrade everything.

That's just my opinion.