The best place to grind covers is the 1st level called "The Bench" You will be fighting 2 lvl 1 thug.
Every time you win you have a chance of winning a cover which will boosts a specific skills for a specific character.
I played the game until I got around to lvl 50, by that time I could win my doing 1 move if it cascaded the longest match I had was 3 turns, the loading screens go by quickly making this the best spot to grind them.
There is no way to know which cover you will get it is random.
To unlock the achievement you need to get 15 for the same character so you need to get 3 skills maxed up on 1 character. If the character only has 2 skills it will not unlock the achievement.
Since the acquisition of the covers is random, having a smaller amount of characters will make it less grindy.
Edit: On second playthrough I started grinding this while my main team was lvl 20 (minimum amounts of characters) and got the achievement in around 1-2hours, went from lvl 20 to 47 with my 3 main characters while grinding this achievement which will make the game go by a bit quicker.
My teams 1st and 2nd playthrough:
P.S. while grinding this achievement on both playthrough I also got these 2 achievements: