Fatal Error

Could you please try to verify integrity of the game, see steps in https://support.steampowered.com/kb_article.php?ref=2037-QEUH-3335.

If that doesn't help, could you please attach/link to your client_log.txt (can be found at Documents\Klei\DoNotStarveTogether\client_log.txt).

I do but not working again http://s4.dosya.tc/server2/v2pm3k/client_log.txt.html dowload my  client_log.txt

The (steamAPI_init() failed) error message stems from within the Steam client API and not the games themselves. Your best bet for getting technical support for Steam client/Steam API issues is to contact Steam Support directly.

In the meantime, here are some troubleshooting steps that you can try that may be of help:

1) Fully exit your Steam client, then fully restart your computer. Once restarted, run Steam once more and see if the games work now.

2) Ensure that you are running Don't Starve as an administrator. Administrator access for PC: http://windows.microsoft.com/en-ca/windows7/how-do-i-run-an-application-once-with-a-full-administrator-access-token

If none of these help, your best, next step is to contact Steam Support (https://support.steampowered.com/kb_article.php?ref=1223-qroc-4460) and inform them of your issue, provide any screenshots of the error message, and they will be able to help get that sorted for you.