So I found the issue. Fluid can't handle namespaces. So first my ViewHelper looked like this:
<php
namespace TYPO3\MyExtension\ViewHelpers\String;
class UpperFirstCharacterViewHelper ...
Now I changed the name of my class and removed the namespace:
<php
class Tx_MyExtension_ViewHelpers_String_UpperFirstCharacterViewHelper ...
This is how it works. At the moment I work with TYPO3 6.1.6. Thank you anyway lorenz for your help!
EDIT:
I went fully retarded! Fluid CAN handle namespaces. I just had to set the namespace the right way. That's how you set the namespace inside the template:
{namespace vh=TYPO3\MyExtension\ViewHelpers}
On top of this comment you can see how my ViewHelper looks like with a namespace.