So in order to override the single pages for the profile section of your site you will need to do a few things...

Copy the single pages and controllers and an element from the core...

/concrete/controllers/profile/* /concrete/single_pages/profile/* /concrete/elements/profile/sidebar.php

To your package

/packages/yourPackage/controllers/profile/* /packages/yourPackage/controllers/single_pages/* /packages/yourPackage/elements/sidebar.php

Then in order to get Concrete5 to realise your trying to override those pages from a package... your going to need to add the following to your package installer and call it from the install method.

//update some core package to be associated with this package to allow overrides within the package
    private function overrideSinglePagePackage($pkg){
        $db = Loader::db();

        /*
        You can use these if you want to override the other core single pages

        $p = Page::getByPath('/login');
        $cID = $p->getCollectionID();
        $db->execute('update Pages set pkgID = ? where cID = ?', array($pkg->pkgID, $cID));

        $p = Page::getByPath('/download_file');
        $cID = $p->getCollectionID();
        $db->execute('update Pages set pkgID = ? where cID = ?', array($pkg->pkgID, $cID));

        $p = Page::getByPath('/register');
        $cID = $p->getCollectionID();
        $db->execute('update Pages set pkgID = ? where cID = ?', array($pkg->pkgID, $cID));
        */

        $p = Page::getByPath('/profile');
        $cID = $p->getCollectionID();
        $db->execute('update Pages set pkgID = ? where cID = ?', array($pkg->pkgID, $cID));

        $p = Page::getByPath('/profile/edit');
        $p->update(array('cName'=>'Personal details'));
        $cID = $p->getCollectionID();
        $db->execute('update Pages set pkgID = ? where cID = ?', array($pkg->pkgID, $cID));

        $p = Page::getByPath('/profile/avatar');
        $p->update(array('cName'=>'Profile picture'));
        $cID = $p->getCollectionID();
        $db->execute('update Pages set pkgID = ? where cID = ?', array($pkg->pkgID, $cID));

        $p = Page::getByPath('/profile/messages');
        $p->update(array('cName'=>'Private messages'));
        $cID = $p->getCollectionID();
        $db->execute('update Pages set pkgID = ? where cID = ?', array($pkg->pkgID, $cID));

        $p = Page::getByPath('/profile/friends');
        $cID = $p->getCollectionID();
        $db->execute('update Pages set pkgID = ? where cID = ?', array($pkg->pkgID, $cID));
    }

You need to be sure that in your uninstall method you set the pkgID back to 0 to reassociate the pages with core.

That should be it... you can now tamper with those pages and re-arrange them as you need to!