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You get a Class C licence by scoring between $2000 and $3000 in Arcade Mode. The good news is that you can do it in whichever time period you wish to choose - so to make this easy, choose the "play for 10 minutes" option. Once you get over $2000, stop playing and let the time run down.

Note that these aren't stackable unfortunately - if you get the Class S achievement, you don't unlock all the lower ranks. So you'll have to do each of these - and the scores on Original Mode - individually.