Assuming you have a Buddypress profile field for your users bio (in this case called ‘Description’), this will work.
function show_description ($atts) {
$user = get_user_by( 'login' , $atts['username'] );
$args = array(
'field' => 'Description',
'user_id' => $user->ID
);
return bp_get_profile_field_data( $args );
}
add_shortcode( 'showdescription' , 'show_description' );