Even with Clarry's excellent screenshots, this is a pain. Here are the steps:
1) Gog.com
2) Drop down the "Your Account" arrow (right of Support) and select Games (3rd option)
3) You will now be on the Games page, where all of your games are listed vertically on the page. In the secondary top menu, select "Orders & Settings"
4) You're now on the Orders page. There's a tertiary left menu here with the first option being "Orders History." Select the second option "Account and Locale."
5) In the page content, "My Identity" options are now displayed. The first one is "Avatar." Select the "Change" button to the right.
6) Upload the image you want and close. There's no save. Changes should be retained.

You forgot one: 7. Wait a day or so for your avatar to change to the new one

Another update sorry for the NECRO:
1)It's ORDERS and settings. No more privacy on the root non-game dropdown.
2)Once you upload and save ONLY on the games page will it display. When you're on the landing page it is just the stock silouhette.

FYI. GOG.. good at hosting game downloads. Horrible at authoring a webpage.
See screenshot for clarity. And of course it shows you as offline. Regardless of your settings.


EDIT: Update: Eventually the game screen shows the avatar.

1. leave your cursor in your profile name
2. click on "Games"
3. then in "Orders & Settings"
4. now click on "Account and Locale"
5. then in "Change" in the "Avatar" section

terrible design