Welcome to Android Central! A few questions:
1. Are you using the alarm from the stock Clock, or are you talking about Reminder alarms from Calendar?
2. In the Clock app, tap the Menu button, then Settings, then Alarm volume. Make sure it isn't all the way down.
3. In the Clock app, under the Alarm Clock section, tap on the alarm that you set up, then check which ringtone is listed--make sure it doesn't say "Silent."
4. When you say the alarm doesn't go off when the Nexus 5 is switched off, do you mean that it's only asleep, or is it fully powered down?