I haven’t seen achievements for winning streak in Hearhstone (any) modes. Maybe you could add such achievements for 5, 10, 15 etc. games won in a row in Hearthstone/arena mode and like 3, 4, 5 etc. games won in a row in battlegrounds and duels ? 
There is essentially that achievement in most modes. Ranked there are ones from every win 3-12. Duels has a 12-0 achievement and arena has 12-0 achievements for every class which is essentially “win 12 in a row”.