You can set the URL for a profile using http://getbraincloud.com/apidocs/apiref/index.html#capi-playerstate-updateuserpictureurl
Uploading an image is a separate issue. You could upload it using the User Files api - but be sure to set the shareable flag to true.
Note - instead of allowing users to upload their own files, most systems these days would allow them to choose from a set of files. You could have a bunch of avatar images available in global files - and set those urls on your player via the UpdateUserPictureUrl() call.