This code works for me.
function mymodule_form_alter(&$form, &$form_state, $form_id) {
global $user;
if ($form_id == 'commerce_checkout_form_checkout' && $user->uid) {
$user = user_load($user->uid);
$loaded_billing = commerce_customer_profile_load($user->field_commerce['und'][0]['profile_id']);
// For each field in customer profile, fill the form field with the $user value.
$form['customer_profile_billing']['commerce_customer_address']['und'][0]['#address'] = $loaded_billing->commerce_customer_address['und'][0];
$form['customer_profile_billing']['field_telephone_number']['und'][0]['value']['#default_value'] = $loaded_billing->field_telephone_number['und'][0]['value'];
}
}
I tried to retrieve the commerce_customer_profile field list, but I didn't find a way to generalize an action for filling every field type in the form.