Please note that I am not an expert like some folks on here.
You might try buying a video card from a local store that will let you return it (I recently bought a video card from Best Buy because they would let me return it if it didn't solve the problem). You would need to open up the computer and determine what kind of slot you have. Just draw the connectors and make note of their size and color and take that with you to the store. If the computer already has a separate video card (versus using a video chip on the mother board), you could take that out and take it with you to the store. However, only do this if you know how to ground yourself (using a wrist strap, or maintaining contact with the metal enclosure). You should also put the video card into a special antistatic bag. Since you don't know if the video card is bad, I wouldn't take it out without these antistatic precautions.
If the video is provided by a chipset on the motherboard, I don't know if not getting a video signal is definitely caused by the video chip or something else on the motherboard.
Oh, just thought of something else. Have you tried a different video cable? Have you checked to see that none of the pins are bent/damaged?
When you open up the computer, check to see that auxiliary power (if any) is connected to the board (Note: not all video boards need auxiliary power connections).
The instructions for the new card should tell you how to ground yourself properly. I would also ask the tech person at the store about this if you have never worked on the inside of a computer before. Electronics are very, very sensitive to any kind of static discharge. Absolutely do not touch a CRT monitor at any time when you have the computer open.
Hope this helps you some.