You cannot do anything after log out .You can write a method and perform all logic and finally return to logout call.
public pagereference logout(){
//perform Business logic here
PageReference pageRef = new PageReference('/default/secur/logout.jsp');
pageRef.setRedirect(true);
return pageRef;
}
Source: salesforce apex callback function onclick of logout in force.com site (http://salesforce.stackexchange.com/q/60990/18731)
check this as well: apex method for logout from force.com site (http://salesforce.stackexchange.com/questions/9499/apex-method-for-logout-from-force-com-site)
use
on click of any button.. It will redirect to logout page..