I would say that you can use the hard drive from your HP but only as a secondary drive and you can use the DVD drive as long as it's a SATA drive. Otherwise, there's no point bothering. I have, however, included both of them in my build.
I've chosen everything from Newegg because I find Amazon hard to navigate for PC components unless you know exactly what you want. I have assumed that you don't need an OS with this but if you do then you can get a worse graphics card and/or monitor to get the $100 copy of Windows 7. So...
AMD Athlon II X2 250 3.0GHz - I have this myself, it's a very snappy CPU and excellent value for money. The included cooler is alright as well.
ASRock AM3 Motherboard - it is a cheap motherboard but on a budget like yours there's really no point in spending anything more on a motherboard seeing as this does everything you need.
G.SKILL 4GB 1333MHz DDR3 Kit - it's the cheapest DDR3 1333MHz 2x2GB kit on Newegg but it has good reviews and G.SKILL is a good company.
HIS HD 5770 1GB - this is a great card and it's a steal at under $150.
Seagate Barracuda 500GB SATA & OCZ ModXStream Pro 500W PSU - a very reliable hard drive with a decent amount of space and a nice, reliable modular power supply with enough power for this system.
Samsung 22x DVD Burner - it's cheap. It plays and burns all formats of DVD.
Rosewill Black ATX Case - on a budget like yours I don't really see any point in purchasing a flashy case or anything. This gets the job done and will have adequate cooling.
Acer 20" Widescreen Monitor - nice looking screen from a well known company with a decent resolution and a good price.
Well, that's what I would recommend for $650 - the total comes to $606.92 which allows you to upgrade something a little if you want (case, monitor, motherboard).
Hope that helped you and feel free to ask any questions.
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