Black screen for about a minute after the startup sequence When I start up my device, I often have to sit through a black screen for a minute or so before it asks me the PIN code of my SIM. This didn't used to be the case before. The black screen comes up after the Nexus' regular loading animation and is completely empty except for the status bar at the top.

Then after entering my PIN I see a 'Process system isn't responding. Do you want to close it?' message on top of my home screen. I can just tap 'OK' and then use my phone like I normally would, but it's annoying.

Why does this happen and how can I avoid the extra waiting time? I have a Nexus 5 with Android Marshmallow.
It may just be because your phone cannot handle marshmallow.

You have a Nexus 5 (released in 2013), therefore when you bought it, it would have been designed for Kitkat. The phone has now been through two other major os updates (lollipop and marshmallow). Because these updates are designed to work fluently with newer models, older phones are expected to see long loading times.