Your avatar (profile picture) represent you across all of Mixer. It's the image others will see on your channel and next to your username in chat.
There are a couple ways to update this:
You can change your avatar via your Broadcast Dashboard's "Customize" page. Simply click on the upload icon withing the "Avatar" section to pick your image, preferable PNG 512x512 pixels.
via Mobile App
You can also change your avatar via the Mixer mobile apps, available on Android and iOS. Log in via your Microsoft account and tap the "Profile" icon on the bottom right, then tap the camera icon to update your account avatar.
I keep getting a message prompting me to link a Microsoft account
In order to upload an avatar, you'll need to have your Mixer account linked to a Microsoft account, you can learn more on how to do this here.
After linking your Microsoft account, you may need to sign out and then back in to upload an avatar.
I uploaded a new avatar, but it didn't change
Some times your avatar may be cached, however it should have changed. Give it a few minutes, refresh and it should show your new image.
If you're on the mobile app, you may need to relaunch the app in order to see the changes on your end.