Well there's good news and bad news.
The Optiplex GX620 has a PCI-Express x16 slot, which is good. That means you can theoretically install any PCI-E x16 card. The bad news is the power supply- the factory psu is only 280 watts, which is too low for most cards with decent gaming performance. Of course you've got an older processor too, but in all likelihood the best card you can install will be a Radeon HD 4650 or GeForce GT220 (which are faster than the Radeon HD 5450). Either of those cards would be a tremendous upgrade from your Intel graphics.
http://search.dell.com/results.aspx?s=gen&c=us&l=en&cs=&k=Dell+OptiPlex+GX620+specifications&cat=sup&x=0&y=0
http://www.newegg.com/Product/Product.aspx?Item=N82E16814102843
http://www.newegg.com/Product/Product.aspx?Item=N82E16814134089
Don't worry about the version numbers being 2.0 or 2.1, that doesn't matter. All PCI-Express x16 cards work in all PCI-Express x16 slots. You just can't fit an x16 card into a shorter x1 slot.
If you had a 300 watt power supply, then faster cards like the GeForce GT240 and Radeon HD 5570 would be recommended. You could still give them a try- they might work, there's just no guarantee.
http://www.newegg.com/Product/Product.aspx?Item=N82E16814133391
http://www.newegg.com/Product/Product.aspx?Item=N82E16814102874
On older Dell computers like your Optiplex and some Dimensions, the power supply is proprietary. You cannot replace it with a regular off-the-shelf ATX power supply.
Performance:
http://www.techspot.com/review/244-ati-radeon-hd-5450/page7.html