Can I purchase Mass Effect 2 DLC without "owning" the video game?

I rent games via GameFly and really want to play the DLC. Do I need to own ME2 to purchase/play the DLC?

You don't need the game to purchase DLC, but you need it to play the downloaded content.

Yes and no. You can purchase some DLC by itself from the Xbox 360 Marketplace, a list of available content is linked here. Most of the important DLC is available in Marketplace.

ME2 also has the Cerberus Network which is a one time purchase for access to the content. Once you are a member, Cerberus Network content is available to you at no extra charge. A list of Cerberus content is here. If you are playing the game through GameFly, the Cerberus network is probably not worth it, however if you do own the game, the addition of Zaeed and the Firewalker pack is probably with it.

ME2 is a massive and rich game, you certaintly would benefit from some of the DLC, but it is not a must buy to get the whole story (with perhaps the exception of the Shadow Broker DLC)