Extract mod contents to the games install directory, the Retail directory as instructed:
/steamapps/common/HITMAN 3/Retail
With use of an application called Protontricks you will be able to modify your wineprefix for the game.
Run Protontricks and select HITMAN 3: 1659040 from the list
You may get a few popup warnings by Winetricks about prefix arch type, but you can ignore these warnings. The next window that pops up will ask you to select which prefix, just make sure it's selected on the default option and hit OK.
Next you want to select Run winecfg and press OK, to then be greeted with the Wine Configuration
Here you want to go to Libraries tab, your DLL overrides list may already be populated, but that's fine, just add these DLL overrides dinput8 and zhmmodsdk, it's lower-case sensitive when you type in the New override for library field box, and you don't need to type the .dll part when adding, just so long the name matches the DLL files. Make sure they're both set to (Native,Builtin), you can check this by clicking on the DLL name in the list and click Edit.
Once you've done all that hit Apply and close everything and launch the game as normal and it should work.