Sometimes, on the PSS, I see that some people's avatars/icons look like Gym Leaders and other NP- characters. How do you change yours to that?
Go to Profile > Click your name > Change avatar/ Public Icon.
That's available after defeating the Champion. There are also apparently locked icons that you will unlock throughout your post-game activities, but they're not fully known yet.
How do people get Gym Leader/E4/Sycamore icons for their PSS?
Unlockable PSS icons?
First, while you are on the PSS click on the icon beside the internet icon (Options), and then click on my profile. Once you have done that, tap your avatar and then the first option that comes up and you can then change it to any thing you want.
Click here for a better explanation