Exactly what it says on the tin, really. Start a new game
(stressed this part as suggestion from Runner7922
, just replaying every level on the Tomb Raider difficulty won't work) with the difficulty set to Tomb Raider, the hardest setting possible.
I would recommend choosing the Tomb Raider hard difficulty level on your first playthrough. As with most games, you'll unlock the achievement for lesser difficulties at the same time (in this case, Complete The Adventure), and as with all Tomb Raider games, the hardest difficulty isn't actually all that harder than lesser difficulties.
The Tomb Raider series is exploration- and puzzle-heavy. As such, the higher difficulty doesn't change a lot to the game. Enemies are a bit tougher, Lara is a bit less tough, and you get a bit less time for saving grabs - that's it. Therefor, even if you usually choose a lesser difficulty for games, I would advise going the hard route on the Tomb Raider series.