1. Download the necessary files from here:
https://www.mediafireXcom/file/jn42h07jk4rmbs5/amral1steam.7z/file
Replace "X" behind mediafire with a dot "."

If you own the Complete Edition on Origin, but wanna play the game on Steam, you can simply use the Alice1 folder from the Origin version.

2. Extract and move the Alice1 folder to:
C:\Program Files (x86)\Steam\steamapps\common\Alice Madness Returns

3. You need to have launched the game at least one time for the next step.
Go to: C:\Users\Administrator\Documents\My Games\Alice Madness Returns\AliceGame\Config
And open the file "AliceEngine.ini" with notepad. Look for "GIsSpecialPCEdition=FALSE" and change it to "GIsSpecialPCEdition=TRUE". This will unlock the Complete Edition content.
Now you will have a main menu option to launch the classic Alice HD game.
Another positive side effect is that all the exclusive Complete Edition costumes and weapons for Alice Madness Returns will be unlocked. This trick was already known and explained in many other guides. However, Steam is missing the HD version of the first Alice and that is the sense of this guide after all.

4. This is the last step. In the same AliceEngine.ini search for "Alice1Path=..\..\..\Alice1\bin". This line is directly above the "GIsSpecialPCEdition=" line. Change it to: "Alice1Path=..\..\Alice1\bin".
Yes, there is a difference in case you wondered. I removed one "..\". This change is neccessary to get the game to run with the Steam version since it has a slightly different folder structure than the Origin/Complete Edition.

Thats it. After this you can launch the Alice HD version anytime from the main menu of Alice Madness Returns.

Have fun playing this old classic!


Other hints and troubleshooting:

I should note that this HD version includes all official languages for the game. English, German, Spanish, French. The game seems to use the language that is used in Alice Madness Returns.

By default the game never launches in fullscreen. You can fix this with editing the default_pc.cfg file in the bin/base directory of Alice1. Just change "seta r_fullscreen "0"" to 1.
You maybe have to delete the other config files in the same directory. I think the only file that you need to have there is the default_pc.cfg. In fact the game puts a config file into your Documents/My Games directory. Just editing this one didnt fix the fullscreen for me though. Weird.

EAX might work. It always reverted to Miles after every restart though. Couldnt fix this sadly. I always changed it to EAX2 in the sound option each start. Or just leave it on default Miles if you dont wanna bother with it. But Windows Vista to 10 shouldnt use EAX correctly anyway. You should try using sound driver tools like Alchemy to enable EAX in the game. Alternatively a tool called DirectSound should work too.

If you have tearing, force Vsync over the driver. The game has no Vsync option sadly so forcing it over the driver is the only way.
And to avoid Aliasing, just use 4x Anti Aliasing in your driver.
I should also note that this game works in 4K. Not kidding, was pretty surprised myself.
Running an old game from 2000 with the Quake 3 engine in 4K is... Well, weird. Back then we mostly played with 800x600 and stuff. Good ol' times, haha.

There is also a nice HD texture pack available from an artist named OmriKoresh. She is very talented! Definitely check it out! This further improves the already "HD" version, since most textures were unchanged and only stuff like widescreen, controller support has been added.
Also, if you play in German, Spanish, French language, there are further patches on the same website to fix some problems that appear when using other languages with the texture pack.
http://www.moddb.com/games/american-mcgees-alice/downloads/american-mcgees-alice-hd

Please tell me if the download file should be no longer downloadable.
You are also welcome to add mirrors in the comments. I will add them to the guide.
Please use 1 single file. So look for a hoster with at least 1GB file size uploads.
Thanks.