At the minute once the user logs in it redirects the to their profile page. How and where do i include that code so the home button disappears?
Those are two different questions. Login With Ajax would help with the redirect. https://wordpress.org/extend/plugins/login-with-ajax/
As for the home button, in your header.php you just wrap is user logged in around Home. See https://wordpress.org/support/topic/367625?replies=5 for a bit more (and some oddities people are having with it…)