Are you looking for a more covert way of killing the tree? Perhaps it is in a neighbours garden and cutting out your light?
Try drilling a small hole (3mm), fairly deep into the trunk. Then, with a straw, pour weedkiller in the hole. Keep doping this for a few weeks and you'll soon kill it.
Don't know why you would want to, but all you need to do is cut and remove a complete ring of bark round the tree. This will kill it from that point upwards as the tree absorbs all its water and nutrients up inside the bark. This stops anything progressing up the tree and it dies - takes a while though.
A dead 'standing' tree can be dangerous and unpredictable as it will become brittle.
Check with your local council to find out if it has a (TPO) tree preservation order on it or not.
Then folow Hawkwalks advice.