How do I change my profile / avatar picture on my account for Starcraft II?

Click on your photo in the upper-right to go to your profile. Then, click on the Rewards menu item, and then Portraits. You can see all of your choices there. You unlock more choices through achievements.

You cannot upload any photo you want.

The old Rewards section in the profile dialog has been replaced with a new Collections tab in the main menu. The Portraits sub-tab allows you to select any portrait you own, which you can activate by pressing the Equip button.

The quickest way I know is to right click on your portrait in the top right corner and choose "Change portrait".