Well if you want to do this yourself first you’ll have to make sure that the form supports enctype:
/** Add Encryption Type to User Forms to allow Uploads **/
function profile_form_edit( ) {
echo ' enctype="multipart/form-data"';
}
add_action( 'user_edit_form_tag' , 'profile_form_edit' );
Then you’re going to have to add a new field that allows users to upload images and supply feedback in the case an image was unable to upload. The easiest way is to grab the image, let WordPress handle the upload through1 various2 functions3, then save the attachment ID into the database ( User1 Meta2 ).
There are a bunch of articles on what metadata is and how to add new usermeta – handling the upload can be done in many different ways so it’s up to on to figure out what works bests. Hopefully that gets your started and good luck!