I just bought Metro since it was 50% off on steam. However when it wont start up, i installed all the steps, but, it wont start up. My graphics card is a ATI Radeon HD 5670.

What error message do you get?
 
Make sure you go through the install folder in steam/metro and install the DirectX / PhysX / C++.
 
I'm also currently struggling to get the game to run, managed to eliminate several errors but more appear, game runs perfectly in window mode, but in full screen I just get a black screen. I think I've remembered why I hate PC gaming.

Did you guys check  the Steam forums?
 
This problem is solved by uninstalling PhysX and then installing it again. Either a restart of Steam after that, or a complete reboot might be necessary. 
 
  http://forums.steampowered.com/forums/showthread.php?t=1305664
 

I got an error message about physx and I have an ATI radeon HD4870.  It disappeared when I restarted my computer and now it works fine.

Ya I had to reinstall PhysX and restart my PC. Seems to be working fine now.