How do I now if I need a new video card?

Yeah download and run Fraps to see how many Frames per seconds do you have when playing. For fast pace action you need at least 60 FPS so you dont lag behind your opponents.
If you are using an onboard GPU I would recomend you to get a graphic car. I am not sure what resolution you are playing at but any upgrade to HD48xx series or to GTX2xx will give you way better performance than what you have now

If you want to be competitive and enjoy better graphics there is little reason to not get a video card unless you are limited via funds.

A $100 card like the 4770 for instance is amazing and will run pretty much anything you throw at mid to low resolutions.

Not so fast. If this is a store-bought computer, the included PSU will be unable to power a high-end video card. Check the amperage available on +12V. The best GPU that a small PSU can handle safely is likely a 4670. That card needs no auxilliary power, yet is vastly superior to any integrated graphics.

Of course, if your budget allows you to replace your PSU too (with a GOOD one, not some lightweight overrated fraud), you can replace your PSU and then get a high-end GPU. You should be able to find a reasonable PSU for around $60, maybe a little less.

Don't feel bad I bought a Quad Core but it came with a crappy Nvidia 8400 video card that was rebadged as a Nvidia 9300. Thinking that I had a low end 9 series I ventured into the online arena thinking I was strapped. I quickly found out that I had entered a gunfight with a butter knife. 12 FPS in COD WAW and UT3 and 60 FPS in Quake 2 !!! Time for an upgrade. First the PSU then the GPU.