Avatars might not be as prominently featured on the Xbox One as they were on the 360's dashboard, but they're still around, and you can use yours to strike a pose and use as your Gamerpic if you don't like any of the pre-made ones on the console.
Press the Xbox button on your controller to open the guide and from there select 'My Profile.' Choose 'Customize Profile' then 'Edit Avatar'. Press the Menu button on your controller and select 'Take a Photo' and you'll be able to switch between Animations and Backgrounds to get the look for your Avatar that you want. Press the View button on your controller to capture the picture and then select 'Save' to make it your gamerpic.