Hello everyone, just to resolve this issue: I had no luck with whatsoever setup.I now just exchanged the HD5770 with a regular PC BIOS HD 6570 (half the price) and it boots without problems. I connect the display to the card, start lightdm via systemctl and get the image.
So it probably was some hardware compatibility or some very kernel specific setting which I am not able to solve (and don’t want to).
Thank you for all your support Johannes