After the son of Death finishes his first day in the family business, he must persuade his father that he wants to follow his dream by becoming a chef.

After Ian hears that his father could be dead, and his horrible uncle was about to take possession of the family property, he desperately needed to find his father.

The duration changes this as its better to rely on the pre existing notion of the angel of Death which you seem to be doing already rather than try and create an original Death character.

This is more of a comedy sketch which means you need a setup and a punch line. Your logline should be structured more like set up the characters involved then present the dilemma and after that a quick pay off.

Perhaps:
After Death's son is made to take over the family business, he must persuade his father to let him pursue his dream of becoming a chef only after killing his first customer he learns to accept his position in life... death.

This could be tied into obesity and heart disease perhaps the greatest modern day killers are the results of Death's son creations.

Hope this helps.

In short, as is the story, my first thought is that he acts like a teenager, he and his father disagree breaking into a full blown arguement. Resulting in death jnr accidently running over and killing the postman, to which death remarks at how proud he is of his son.