Just like it says, complete the 10 chapters of the game's campaign on extreme difficulty, this can be done with any combination between single player and co-op. To actually unlock the extreme difficulty you have to play through the game atleast once first. The game does get frustrating now, so keep your cool, be patient when picking off the enemies and try no to rush the level.
Side Note: These achievements should actually stack up, so completing the game on the hardest difficulty should also unlock the achievements for the lower difficulties.
The 10 missions to be completed are 'Santa Cecillia Monastery', 'Sveta Ostrov Whaling Station', 'Kyrkalov Submarine Station', 'Kolyma Castle', 'Metuwe Township', 'ATAK Diamond Mine', 'Chemical Tanker', 'Salem Union Sawmill', Petro Naviera Refinery' and 'Military Command Base'.
You can see which missions you need to complete by clicking on 'Awards' at the main menu then 'Statistics' and the pressing
twice. A list should then come up with coloured stars next to each mission to verify which mission has been completed and on what difficulty.