BuddyPress.org

Look at /bp-xprofile/bp-xprofile-signup.php. I’m not writing your code for you, but from the looks of things, I think you need to remove_action() the xprofile_add_signup_fields() call, and add_action your own version of this function, but obviously removing the avatar form field.

Best place for the code would probably be /plugins/bp-custom.php and make your add_action have a priority of 11.