I cannot play my batman arkham city dlc's, help?

First of all, make sure steam says it installed them. Sometimes with games steam will download, and then when you hover over it in your steam library it will say "install" where "play game" should be. This normally only happens if a new version was released while you were still downloading the previous version.

If steam claims to have downloaded and installed them, then it should be working. One thing to keep in mind, is that the robin/nightwing and harley dlc is not like the catwoman dlc. At points in the story, you will stop playing as batman, and play as catwoman for a while, but the other dlc packs don't do that. To use the robin/nightwing, you need to swap to them at the character swap points, or from the main menu go to the riddlers revenge. It should show 4 characters (batman, catwoman, robin, nightwing) on the selection screen. Harley quinn's revenge is also only accessible form the main menu (should have a little icon with harleys face). If that still does not work, then contact steam customer support.

you have to buy the game of the year edition in order to play as robin and to have harley quins revenge

make sure they have downloaded proply