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.