First of all, you need to check the power supply. If it not over 300W I would not even consider an upgrade. And even that, most 300W power supplies found in factory build computers make theirs cheaper by putting some sort of generic PSU in the computer that does not perform up to par with the class leaders.
Next, you may want to consider your system in general. If it has things such as 1GB of RAM or a Pentuim D processor, those things would definitely bottleneck the graphics card from performing at its best. An easy and effective upgrade to perform is an upgrade to the RAM itslef, it causes a noticeable upgrade. Make sure you are buying the right RAM sticks though, there are different kinds.
Also, consider your budget. one of the best for under $100 is the Radeon 5670 (6670) as it gives good performance and doesn't break the bank or use too much power. Yet a PCI-E x16 1.0 is rather ancient. Though the computer is slow enough itself that it will most likely not be an issue