This is very easy to get once you have some money, and can be done straight after the first area if you wish. Once you get some cash from selling items you can buy upgrades on the shop terminal to the right of the computer. One of these options is the axe. Once you've purchased it, press
to equip it, and interact with a customer with
to unlock the achievement. You can do it to Uncle Billy too if you don't want to wait for a customer (thanks to Jayadub for confirming).