Where can I check what Xbox 360 console is linked to my Windows Live account?

As far as I know, there's no direct way of getting this information. However, you can safely go ahead and make your purchase anyway. Since downloads are associated with your account, they'll appear in the Active Downloads section of the Marketplace pane on the Xbox button menu no matter which Xbox 360 you log in from.

If the current Xbox 360 is not considered your primary console, the item in the queue might indicate that it's "Assigned to other console." If this happens, selecting the item and pressing A should bring up an option to download all items. Choosing to do so should update your account to consider the current system as your primary Xbox 360, and downloads will happen automatically when you log in from that point forward.

Microsoft has a Knowledgebase Article on this topic that might also contain relevant information, and Major Nelson did a blog post when this first became a feature.

I'm not sure if it's necessary to do it manually if you go through the above step, but you may also need to transfer licenses for any content you had originally downloaded on your other console.

To perform a license transfer, on your console, go to the Settings pane. From there, click on Account, and select the License Transfer option. As you can only perform this action every four months, it'll ask you to confirm. Doing so should transfer all of your existing licenses over to your new Xbox 360.