Firstly - the desktop shortcut (which you are undoubtedly using since you are here) will only take you to the base AoE3 game, with no expansion content. To play the expansion content, you must open the game via your game library in the steam window. Select Age of Empires 3 Complete Edition and then click play, and a drop down menu will appear asking you which expansion you want to play. Done.

Some users have also stated that they had to open steam as an administrator before the DLC was activated/unlocked for play. Though this has not been the case for all players, if you still cannot get The Warchiefs and The Asian Dynasties to work using the above method, try running steam as an administrator by right clicking on your Steam.exe shortcut and selecting 'run as administrator'. Good Luck.

P.S. I recommend choosing the Asian Dynasties expansion, as this is the lastest and thus includes all previous DLC content (eg The Warchiefs).