"Keep on cleaning."
Translation:Continuez à nettoyer.
You need to conjugate the first verb. Continuer is the infinitive form of the verb. "Continue à nettoyer" for "Tu form" or "Continuez à nettoyer" for the "Vous form". Only the first verb is conjugated - subsequent verbs are in the infinitive. The verb continuer uses the prepositions "à" or "de" (most use one or the other).
"Garder" does translate "keep," as in "gardez vos chaussures" ("keep your shoes (on)"), but I believe it is an error to give it as a translation for "keep on" (though if you reword my example sentence as "keep on your shoes," you can see that one could argue that it's a valid hint). In Québécois French, there would be no difference. The main differences between Québécois French and French French lie in anglicisms (both use them, but they use different ones and the Québécois use more), slang, and terminology for modern technology.