Create a new button on the Contact object (Display Type: Detail Page Button, Behaviour: Execute JavaScript) and put this javascript on it:
{!REQUIRESCRIPT("/soap/ajax/20.0/connection.js")}
var postcode = '{!Contact.MailingPostalCode}';
if (postcode != '') {
postcode = postcode.replace(' ','%20');
sforce.connection.remoteFunction({
url : 'http://api.postcodes.io/postcodes/'+postcode+'/validate',
async : true,
cache : false,
method: "GET",
onSuccess : function(response) {
if (response.slice(-5) == 'true}') {
alert('OK');
} else {
alert('Wrong - check using https://www.royalmail.com/find-a-postcode');
}
},
onFailure : function(response) { alert('ERROR: '+response); }
});
}
For the Account object use {!Account.BillingPostalCode}
or
{!Account.ShippingPostalCode}
as appropriate.