Will the 4850 fry my cheap hp computer?

i would say that you get a radeon 4870 because of the fact that the fan exausts the heat outside the case but can your psu handle it?

You should be fine for temps.
If you can, screw/ziptie/tie a 120mm fan inside the grate blowing fresh air on your system.

I would be more woried about the PSU.
Did you plan on upgrading the stock HP PSU?
If not, you really should.
OEM PSU's are known to be very low end and your new 4850 is going to draw some power...

I'd do the 4870 but I think it would be gimped heavily by my e2200 (2.2Gz dual core). The 4850 will, I'm sure, be somewhat gimped.

Even bottlenecked bye your CPU a faster GPU will let you turn up more eye candy with no performance drop.

If you want to get saucy, you could always Pin Mod your E2200 to 2.926Ghz

Of course, dasickninja's disclamer goes here...

Disclaimer:

Be aware that this mod will void your warranty like no ones business, especially if you aren't precise in you work. Before attempting this, make sure you have the orientation of the chip correct, and you know which pins you are connecting.

EDIT:
I keep forgeting Toms will not let you link to that site.
If you really want to know, check the CPU section at

X
C
P
U
S.com

If you are concerned about temps, I think what I'd do in your case, get a PCI slot cooler fan, and mount it directly under your video card, or above it depending on the orientation of the case, I know some newer dells open from the opposite side, so like when I mounted a video card in one at work, the fan was actually facing upward, not downward.

Also, another thing you can do for heat, take out the stock exhaust fan, and get another one that can flow more air, but is the same size. Also, they used to have fans that could be mounted in an extra cd/dvd drive tray. Or you can rig up a high flow 80 mm one in your drive bay openings using zip ties or something similar. If you can't see inside the case, whatever you gotta do, if you "rig it", if nobody sees, who cares.

Just keep an eye on temps. It's not hard to add a fan or like Ohio said use a PCI slot cooler fan to help. The upgrade in graphics is worth a little extra heat.

+1 4850 with pci slot cooler dirtmountain mentioned.

someone mentioned a new psu. you will need more than 450w, regardless of min requirments. Fans, opt drives, hard drives, usb devices ect will add to the total system wattage.

Get a 550w or higher.