In order to unlock all abilities on a character, you need to be Class 4 (so all the abilities are available) and level 99 (to have enough skill points to unlock them all). Each character has their own achievement for unlocking Class 4, so I'm not going to go into that here.
I highly, highly recommend that you do this on a New Game Plus playthrough, since you get an ability that gives +300% experience upon starting NG+. You can also choose one of the same characters from your first playthrough and keep their level (their class gets reset, so you will still have to do everything to unlock class 4).
The 300% boost, combined with Lil Cactus and my 60+ levels from the first playthrough meant that I hit level 99 before even unlocking Class 4 and doing zero extra grinding. As soon as I re-unlocked the final class, I dumped all the points and got the achievement.
You could still do this on the first playthrough of you really don't want to touch NG+ but you might go insane from the grind. Your choice.