Updated April 19, 2017: The Xbox beta app actually allows anyone — not just Xbox Insiders — to upload custom gamer pics. The article has been updated to reflect this.
The ability to set a custom gamer pic recently started rolling out to Xbox Insiders in the Alpha ring, but it has thus far relied on a somewhat clumsy process of transferring pictures to your console using a USB thumb drive. Thankfully, Microsoft is easing that process a bit while opening it up to everyone by allowing custom gamer pic uploads directly from the Xbox beta app for Windows 10.
Uploads from the Xbox beta app should make for a much more convenient process. To get started, all you have to do is make your way to your profile page from the hamburger menu. From there, click "Customize" and then select the "Edit" option next to your gamer pic. Upload a custom picture, save it, and it will be applied after passing through a short review process.
Keep in mind that any custom picture you want to use must meet the 1080x1080 minimum size restriction. And if the option to upload a custom picture isn't showing up in your app, restarting it should fix the problem. If you don't count yourself among the rather limited group of Xbox Insiders on the Alpha ring, you can grab the Xbox beta app now and start toying around with your own custom gamer pics.