I don't claim to know everything about upgrading, but it seems that if you're going to be upgrading a microATX HP computer like that, you're going to want a better card, since you won't be able to add a second for crossfire/SLI later on.
My suggestions would be either the:
GTX 460 (768MB) - $99 (with rebate)
GTX 460 (1GB) - $140 (with rebate)
The GTX460 outclass the HD5670 in every way, and only for an additional $10! The 768MB version would be more than enough for what you're thinking of getting, and it'll most likely be able to run MW3 when it's released. If you can afford it, you should spring for the 1GB version though. They're also DX11 compatible, which is good for future games as well.