Magento 1.9 – How to Redirect to Controller Action from phtml

You get redirected because you call parent::preDispatch() in your code.
This calls the initial method and it does not pass the validation because you action is not in the list of allowed actions

    $openActions = array(
        'create',
        'login',
        'logoutsuccess',
        'forgotpassword',
        'forgotpasswordpost',
        'resetpassword',
        'resetpasswordpost',
        'confirm',
        'confirmation'
    );

But why do you need to rewrite the default account controller? Can't you just have your own controller that does not depend on the functionality of the default one? You controller should only contain ajaxloginPostAction() that does what you need. I assume it should be something like the loginPostAction does but returns the response as json.
It should work in theory.

add in your confix.xml in

<frontend>
            <routers>
                <customer>
                    <args>
                        <modules>
                            <company_modulename before="Mage_Customer">Company_Modulename</company_modulename>
                        </modules>
                    </args>
                </customer>
            </routers>
        </frontend>

these is you controller

require_once 'Mage/Customer/controllers/AccountController.php';
class Company_Modulename_AccountController extends Mage_Customer_AccountController
{
    public function logoutAction()
{
    $this->_getSession()->logout()
        ->renewSession();
      //add your code here
    $this->_redirect('*/*/logoutSuccess');
}
}