How to enable push to talk for Skype for Business
"Push to talk" is currently only available in "Skype". "Skype for Business" is currently lacking this feature. You can however press "Windows logo key"+F4 to toggle mute on/off. It's a global hotkey, so you don't need to have the window focused. When you mute the mic you will hear a *diu* sound and the mic icon will also change into muted mic icon. When you unmute there's no sound, but the mic icon will change back to into mic open icon.
At time of writing this, Skype for Business does not have a native push to talk feature.
There is a feature request for this here.