Several playthroughs needed, each license needs to be separately done in one game.
You'll need to following amount of money to reach each license
C class license $2000 - $2.999
B class license $3000 - $3.999
A class license $4000 - $4.999
S class license $5000 - $9.999
If you go over $9.999 you'll gain the Awesome and even Crazy! Licenses (if you go over $20.000).
None of them will give you an achievement, not even the S class license achievement. The game probably thinks you're too good for the S class achievement if you can get higher ratings. To gain an achievement your total should be between the set amount of that license, i.e. between $5000 - $9.999 will get you the S class license but if you already have that achievement you won't get the A class license or lower if you get that "score" again.
You could play by arcade rules to gain the achievements but it's faster and easier to play 10, 5 and 3 minute games to get each achievement. Four 10 minute games are more than enough to get the S and A class licenses in both modes, Two 5 Minute games are enough to get both B class licenses and for the final 2 C licenses you can play a 3 minute game.
The only thing you need to know is that you can get extra money by performing crazy jumps, crazy drifts and near misses but it's hardly worth the risk because in 10 minutes you should have no problem getting more than $5000 for a S class license.