The requirements for the games you plan are very low.
The ATI Radeon 9600, Pentium 4 and AMD Athlon XP are very dated components now. You DO NOT want to be buying them.
The monitor resolution that you chose (1024x768) is very small by today's standards. Unless you plan on using an older monitor which only supports that resolution as its highest, consider buying a monitor with a resolution of at least 1680x1050. It will definitely be a plus for gaming (some games may not even run at 1024x768).
Take a look at some of the builds that Tom's have done. For $1000 you can get a fairly powerful build for gaming, much more powerful than you need for the games you have listed above.
If you are trying to get it under $1000 WITH a decent monitor, the $500-$600 are probably better to look at.
Both are more than enough to play the above games and still leaves you about $500 to add a monitor and any other stuff you need to buy.
There are also the $1000 builds, which add the Intel Core i5-2500k to mix, which is easily the best for value gaming processor on the market. I know you prefer AMD (I do as well XD) but its undeniable that the i5-2500k is the king at the moment. In my opinion, unless you plan on playing games that have higher requirements in future (say, Crysis 2 or Skyrim), the $1000 build is way too much for you. Here they are anyway:
Finally, to answer your question, buy or build?
Build will save you money and allow you to get a more powerful computer for a cheaper price but it does require you to do a bit of research on how to put it together. It's not hard, almost like putting together a LEGO kit, just much more expensive =P
Taking a look at videos on Youtube can help and there are also sites which have step by step guides. Here is one guide: http://www.tomshardware.com/forum/274745-31-step-step-g...
On the other hand, buying guarantees that everything works (and if it doesn't you can always return it to wherever you purchased from) and its already put together for you. But you will pay extra for this service, as opposed to buying components and building yourself.
EDIT: Disregard most of the above =P
aznshinobi is right, Llano is enough for your needs.