Google Nexus 5 doesn't boot
I suggest you if you still have warranty to bring it back to the store.
However you maybe should read this first:
Nexus 5 and Nexus 7 Lollipop Update Brick.

The Nexus 5 and the Nexus 7 can be bricked in lollipop, and it only can boot to the Google logo. I don't know if this is your problem but it may be a part of your problem. If you update your phone to Android 5.1 then it should be fixed.