Chase is more general: in a game of football, the players chase after the ball (ie to get close to it and seize control of it in order to pass to their teammates), the police chase after criminals to catch them and arrest them. The chaser typically can already see who the person he is chasing. Hunt is more aggressive: it suggests a predator / prey relationship. The hunter probably cannot see his prey. To hunt animals to kill for example. But also it is used figuratively: "I am going to the shops to hut for a new coat"

Hunting usually means the practice of killing wild animals for sport or food, think hiding in the bushes with a gun, or shooting birds etc. Can also be used as in house-hunting - looking for somewhere to live. And also "hunting for" something, a job etc. Chasing more generally means the pursuit of something or someone. "He chased me around the house". To complicate matters though, "the chase" refers specifically to hunting - "she liked to follow the chase". Hope that helps a little!