To Obtain a A-Rank, you need to do the following
Finish the Boss in less of 3 Minutes
Have 3 Health during the fight
Parry 3 Times
Use 6 Super Meter Cards (I recommend using all 5 for the Super Art and 1 more for the EX)
And have a Skill Level of 2 Stars
You don't need to do every single thing, you can finish the fight in 1HP and as long as you did everything else in full, you'll get a A Rank, here's a guide for all the bosses in Inkwell Isle 4