Is there an easy way to figure out which DLCs have you bought already?
Solution 1:
As of the Steam Discovery Update on Sept. 23rd, 2014, Steam itself now shows you the DLC you own in the Store page without having to use browser extensions or look at the game properties in your library.
The blue markings on the left side of each DLC entry indicate that I own it already and if I hover over it, there will be a popout that shows that it's in my library.
Note that isn't perfect. In this particular example, Steam shows that I don't own Wonders of the Ancient World, even though I do, but I'm guessing that's a problem with that particular piece of DLC, as it's done that since release. However, Steam still won't let you buy anything you already own, so you're safe from double purchases.
Solution 2:
Steam has now implemented a menu to allow you to view the collection of DLC you own.
Right click the item in your library, and select the "View Downloadable Content" option:
This will bring up a window in the game's options, listing the DLC you have purchased for the game:
Solution 3:
Others already stated the direct and true answer. Let me tell you about a plugin that makes these steps obsolete.
With Enhanced Steam, a browser plugin, you can see which games you already bought (or have in wishlist) while browsing the Steam store. While in a list of games, all bought games and/or DLCs are coloured green so you can easily see what you already have got.
It has also more features, like comparing prices in other regions, listing sales for a product or warning of 3rd person DRM. You can see this in the features list.
It is one of my favourite plugins, as it really enhances my experience on the Steam website.
Here is an example of some DLCs the guy already bought:
Solution 4:
The best thing I could find is by logging into steam though their website And go to your account information.
There you can see all of the items you have purchased in the past, including DLC.
As far as the actual Steam Client goes, the best you can do is right click on each game that you have and select "View Downloadable Content" but that is not as comprehensive of a list as going to the website is.
Solution 5:
There's a Chrome extension called "Enhanced Steam." This will highlight games and DLC you already own in green. It only works on the web version of the store. It will also highlight if you have a copy of a game in your inventory, if any of your friends have it on their wishlist, what price its selling for elsewhere, etc.