how is he sending the threatening messages? I assume by text, email, phone or other form of media? If so he is committing an offence under telecommunications legislation, aside from him committing the common law crime of threats. For that to occur the perpetrator must be in a position to carry out the threat being made, and the victim must genuinely perceive the threats to be genuine and be placed in a state of fear or alarm. In other words if the threat he issues is suggesting he will drop a nuclear bomb on your house, well we all know that threat cannot be carried out. If however he says he is going to stab you and he is known to be violent then that threat could be perceived as a serious one.
Retain all messages he sends you, show them to the police as he is committing an offence. As you are the complainer and it is not a serious crime, the police should proceed as you ask, so either ask for it to be dealt with formally and make a complaint, provide a statement and be prepared to go to court. Or ask them to simply warn him to stop or it will be dealt with formally.