For all those facing the same problem, its actually pretty easy. Log into the steam backend and set up an achievement like so:
Then in leadwerks, if you use lua, just call Steamworks:Initialize() in the main.lua at start. To set an achievement active do this:
Steamworks:SetAchievement("STORY_COMPLETE")
--And thats all :)
Two things to keep in mind and which confused me are:
1. Steam may take a while to display your achievements after you published them, so wait until you see them displaying in the steam store, or when you click on the game in your library.
2. The achievement unlocked popup, will NOT show up while ingame, you only see after you closed the game and an achievement has been unlocked.
There are also progressive achievements but the work pretty similar.
Little tip: As those functions are not documented, activate auto complete for the Leadwerks IDE and type "Steamworks:" it will display all available functions.
Good Luck
I fixed the problem on my own now. After disabling various things and doing the map change in main loop I could nail down the problem. It was caused by many diffrent sources and turned out to be pretty complex, still could fix it luckily. Thanks for all who posted, your information was helpful indeed.e
Phodex Games's post in How to set a Steam Achievement was marked as the answer