There is no clear description in the game on how to get each award. Most of my guide is made out of my experience playing the game or my assumptions. If you have anything to say, please, do so in the comment section.
Note that this guide contains STORY SPOILERS.

The Graduate
- Finish the tutorial (don't skip it or it will not count!)


The Past Unearthed
- In "The Wrong Ditch" learn everything about Richard Hannay's past


Second Life
- Progress through the story. (You will get this one while meeting Scudder)


No Stone Unturned
- *SPOILER WARNING* Explore every evidence in the room with Scudder's dead body


On the Run
- Progress through the story (You will get this one after jumping on the train)


Killer Stories
- While on train, read about the murder in newspapers (second page)


Sudden Inspiration
- Progress through the story (You will get this one while staying at the Inn)


Nothing But the Truth
- Decode Scudder's book and read every page


Unbelievable Luck
- Progress through the story (you will get this one while escaping Black Stone)


Demolition Man
- Explode the wall with lantenite without doing a mistake
(you can maybe do one mistake, I'm not so sure)

Full Recovery
- Progress through the story (You will get this one after recovering your shoulder and legs)


The 39 Steps
- Progress through the story (You will get this one after discovering the meaning of the phrase "the 39 steps")

I know Germans
- Progress through the story


Ultimate Explorer
- Explore every location in game
[Make sure in chapter "Desperate Measures" to check both windows and locked chest BEFORE you find lantenite in the wardrobe. If you still have trouble unlocking the award, make sure you also watch the credits until the end (Credit: Weffke).]

Know It All
- Examine every item/evidence in game (not so sure about this one, I got it immediately after I beat the game)

Final Word
- Watch the credits and final scene till the end

Now your Award screen should look like this: