hunts down a Criminal in order to kill him or the Main Character is the Criminal and has really killed someone and is being hunt down by whoever

The japanese drama Guilty Akuma to Keiyakushita Onna should be what you're looking for. But also a few revenge period dramas. Right now, I can only think of Strongest Chil Woo.
Ouroboros (fits both requests)

a movie if at all interested: Brain Man (he hunts down killers and kills them, and cops are chasing him because he kills, so it fits both requests)
Death Note (jdrama) = main chara wants to hunt down his opponent, opponent wants to hunt down main chara, main chara is actually evil and therefore wants to kill his opponent

Bloody Monday (jdrama) = main character is hacker who has to hunt down criminal, it's harder than it looks tho because criminal is intelligent

Maou (jdrama) = main chara is police and wants to hunt down criminal not knowing that criminal seeks revenge, this is the best revenge drama I've ever watched

Majo Saban (jdrama) = a trial, it's unclear what happened and who the real victim is and who the criminal, this was full of twists and unexpected ending


Ryusei no Kizuna ---> this is surprisingly funny as well
The gir who sees smells (kdrama)
Triangle (jdrama)
Joker (jdrama)
Byakuyakou (jdrama)