This can be done with the Silent. There are several cards that apply poison to enemies:
There are also a few relics and a potion that can help, so keep an eye out for those as well.
I was able to easily get an enemy to 99 poison using these two cards:
When upgraded, Catalyst TRIPLES an enemy's poison, and with Burst it would instead triple it, then triple it again. With these two cards alone you'd only need an enemy to have 11 poison (11*3=33, 33*3=99). Or if you also had Burst upgraded, it would play your next two skills twice, so you could play, say, upgraded Deadly Poison for 7*2=14 poison, then Catalyst for 14*3=42, 42*3=126.
Since poison damages an enemy each turn, you can't build it up slowly or the enemy will almost surely die before you get to 99, making it so Catalyst with huge buildup in one turn is the way to go.