Basically the 5870 is like the 4870 except it has support for Direct X 11.
The 5870 is a bit faster than the 4870 when it comes to direct X 11, a nice card.
ATI cards run extremely hot though, you need to make sure it doesnt overheat, and seeing the inside of your case, it looks like it might not have enough room/air.
Also think about upgrading your motherboard/CPU/Ram
Eh, well I just want to get a decent card that can play the latest games and that i just have to plug it in there, install and i'm done(without upgrading anything else). any suggestions?
Edit: Actually sorry. Your power supply is only 300 Watts and will not be able to power any higher end cards. You need at least a 500 watt power supply with either pci express connectors or a whole bunch of molex connectors.
If you try to use your 300 watt you might fry your motherboard or something, since your components wont have enough power.
500 Watt power supplies are kind of cheap so I bet you can go to best buy or something and get one. Make sure you get a decent brand one, because alot of the cheaper brands die really fast and arent sufficient for powering graphics cards.
Upgrade that PSU ASAP
Once you upgrade that power supply you can get any graphics card you want, use this mentality: The more expensive the better
I highly recommend the 5870*(typo). Once you upgrade both your PSU and graphics your system will be destroying games.