Hey @Greubisan,
did you try to run it as admin or reboot your System.
If you have client logs send them in this thread please (%appdata%\TS3Client\logs).
Try to make delet the %appdata% folder from ts3 and then install it again.
or try the fixes in this thread Teamspeak 3 Client Not Starting (MSVCP140.dll , VCRUNTIME.140.dll) - #7 by otoroot
Cheers 
