Do I need to run the VPN client on both my phone(hotspot) and on my laptop(or connected device)? Or just one or the other? If just one which is recommended?
You only need to run it on one. If you use vpn on your phone when not tethering then just leave it on when you connect the laptop.
Just do a quick test to make sure its working by tethering to the phone with vpn off and checking your ip on the laptop (https://whoer.net/) then turning vpn on, on the phone and refreshing whoer on the laptop making sure the ip changes.
It is not naturally possible to share a VPN connection over a hotspot. Android software prevents other devices from sharing its VPN tunnel, so if you connect your Android device to a VPN, and then share the internet connection to another device with a hotspot, the other device will not be routed through the same VPN tunnel as your Android device. Instead, it will use your native ISP connection.
To potentially achieve this, you need to root your device which is a risky process if you are not experienced with it, and we do not advise our users to do that.