I've finally managed to make it work!
The GetResource looks like this:
[[getResources? &depth=`0` &includeTVs=`1` &processTVs=`1` &includeTVList=`TV_Article_Feature_Image,TV_Article_Feature_Image_AltTag` &processTVList`H4HAU_TV_Article_Feature_Image,TV_Article_Feature_Image_AltTag` &tpl=`Chunk_Intro_InfoTpl`&sortby=`{"menuindex":"ASC"}`]]
The Chunk_Intro_InfoTpl chunk looks like this:
<div class=artlogo><a href="[[~[[+id]]]]">[[+tv.TV_Article_Feature_Image]]</a></div><div style="margin-left:168px;">
<h3 style="line-height:1.2em"><a href="[[~[[+id]]]]">[[+pagetitle]]</a></h3>
<p>[[+introtext]]</p>
</div>
The "Output Option":Alternate Tag: of the TV_Article_Feature_Image looks like this "[[+tv.TV_Article_Feature_Image_AltTag]]"
The +tv.TV_Article_Feature_Image_AltTag is probaly what made it worked! Is processTVs the setting that makes it turn the TV_Article_Feature_Image_AltTag into actual text the the user entered?