Here are my recomendations for each.I kinda went very high quality parts and decent price in your budget.The monitor would be one of your choice,for the information you got me i would need to know what video card you got and whats your current monitor to consider an upgrade.
-Case:Lian Li PC-7B plus II. No fancy lighting,no fancy colors just a no nonsense aluminum case with great airflow. I got a similar one and i can tell you theres no other case at this price range that beat it.Other cases would be the Antec 900,Coolermaster RC 690,LIAN LI PC-A05B.
http://www.newegg.com/Product/Product.aspx?Item=N82E16811112099
-Power Supply

C Power & Cooling's Silencer 610 or Corsair 550 VX
http://www.newegg.com/Product/Product.aspx?Item=N82E16817703005
http://www.newegg.com/Product/Product.aspx?Item=N82E16817139004
-RAM: Corsair Dominator DDR2 1066
http://www.newegg.com/Product/Product.aspx?Item=N82E16820145043
-Motherboard: GIGABYTE GA-P35-DS3L
http://www.newegg.com/Product/Product.aspx?Item=N82E16813128059
-CPU: Intel Core 2 Duo E6550.
http://www.newegg.com/Product/Product.aspx?Item=N82E16819115030
TOTAL: $553.96
The Heatsink that comes with the Intel CPU is good but if you are gonna overclock get a Coolermaster Hyper 212. It should help you reach around 3 GHz speeds on the E6550. I got a E6420 @ 3.1GHz running for months with that heatsink and i couldnt be happier.
Forget about Barebones PC's and cases with PSU's included.most of the time they got pretty crappy components so they can be sold cheaper,that give rise to problems later on.
Also don't forget if you are gonna be using Windows you gonna need a new OS cd,the one from the HP wont work. Unless you got a retail edition of windows,you gonna need to buy a new one.OEM versions are cheaper that Retail but can only be used on one Motherboard(Vista OEM authentication lets use it on your next motherboard i think) unless it breaks down.