Check the documentation on your adapter to make sure that it is designed to convert a VGA input from the computer to an HDMI output, rather than the other way around. VGA to HDMI and HDMI to VGA are two different things (though visually you probably can't tell them apart), and they don't necessarily show the right one when doing internet searches, so you might have ended up with the wrong one if you did a google search and simply bought from a link that popped up. You also won't get audio to your tv unless the adapter includes a connection to the computer's audio output, since VGA doesn't carry audio.
Also, check to see what type of expansion slots you have available for your computer. Your current video card is likely a PCIe card, since that standard came out in 2002. You could replace your card with a newer one with an hdmi output, though you'd want to make sure that drivers are available for your operating system, since it likely hasn't been updated to the current versions being used today. The good news is that you wouldn't need a current generation card, and can get older ones for probably $80 or less that should work with your system. If you don't have PCIe, then you will have a harder time finding a card with HDMI, or possibly finding a card at all.
Mike