There's two major points that need to be addressed, I think, and possibly a minor one as well.
One is the physical aspect. That pages suggests you should have a single PCI-e x16 expansion slot, which may or may not be half-height. Breaking that down: single is fine, as you only need the one video card. PCI-e x16 is the interface (type and size of slot), and is good - older systems used AGP for video cards, and that would have caused problems, but PCI-e x16 has been the standard for a few years, and will likely be for a while, too.
Half-height is an issue, as the card you linked (and most decently powerful cards) are full height - they're "taller" and stick out further from the motherboard than the case will allow. Of the three computers pictured on that page, which does yours look like? Only the largest of the three supports full height cards.
The second issue is power. Video cards can use a lot of it, and your PC doesn't have a very beefy power supply at all - it varies between chassis, but none of the ones listed there top 265W. In addition, I strongly suspect it doesn't have any dedicated PCI-e power connectors (6 or 8 pin plugs specifically for video cards, to meet the power demands of higher end ones). AMD suggests a 450W PSU for that video card (although that's a very conservative number and you could likely get away with less), and it's going to need a 6 pin PCI-e power connection.
Normally, I'd suggest replacing the stock power supply with a somewhat beefier unit from a reputable manufacture. However, some Googling suggests Dell is using a non-standard power supply that is a little taller than the ATX spec ones. Because the Dell units are bigger, if you're willing to drill a new screw hole or two you could replace likely it, but that's starting to get kind of involved. If you want to go that route,
this is what I'd start by looking at.
A simpler option would be something like
this card*, which is a little cheaper, will fit a half-height case, and doesn't need as much power (or a dedicated power connector). It'd still be pushing your power supply more than I, personally, would really like to see, but it would be well within spec. On the other hand, it's notably slower - probably only about 60% as fast as the 5770.
That's brings me to the minor point - depending on the resolution you're playing at, even the 5770 getting a little slow for the more demanding of new games at good settings. With a 720p or even 900p display, you'd probably be just fine, but 1080p or greater and you might be better served spending a little more for something slightly faster still - a GTX 460 can be had for ~$25 more and is 20-30% faster. Of course, even the 6670 would absolutely crush what you have now, and you'd at least be playing those games, you just might have to skimp on the eye candy at times.
*
Not sure this one actually includes the low profile bracket. This one does, and is even cheaper, but is out of stock at the moment