Please post your error on the dev site, Steam itself doesn't offer technical support for Dota 2: dev.dota2.com.
Now my 2 cents: it misses a dll-file, maybe you need to check your Direct X, I had some similar issues, when PES didn't want to start, because the installer failed to install the newest Direct X.
after some hard and serious investigation on Google, i found a fix:
go to your steam folder: Steam/SteamApps/[LoginName]/Dota 2
rename bin into bin_old
make new folder: bin
start the game
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I have a problem with Dota2 so if someone can help me
http://i48.tinypic.com/2b5sub.jpg
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