This “How To” assumes you have installed both WHMCS and Kayako.

You need to copy the files from your Kayako SupportSuite install
Located under:
public_html/supportsuite/integrate/whmcs/modules/support/
copy the entire /support/ folder and paste under
public_html/whmcs/modules

When done you should have
public_html/whmcs/modules/support/kayako/
announcements.php
downloads.php
knowledgebase.php
submitticket.php
supporttickets.php
viewticket.php

and

public_html/supportsuite/integrate/whmcs/templates/kayako/
copy the entire /kayako/ folder and paste under
public_html/whmcs/templates

When done you should have
public_html/whmcs/templates/kayako/
announcements.tpl
downloads.tpl
knowledgebase.tpl
submitticket.tpl
supporttickets.tpl
viewticket.tpl

Open
public_html/whmcs/modules/support/kayako/integrate.php
and make the required changes.

For WHMCS 4(and now confirmed on WHMCS 5) thanks to Matt from WHMCS for providing this fix. You need to open:
modules/support/kayako/integrate.php and change:
if (!empty($GLOBALS['clientsdetails']['id']))
{
require_once ("./includes/clientfunctions.php");
$GLOBALS['whmcsuser'] = getclientsdetails($GLOBALS['clientsdetails']['id']);
to

if (!empty($_SESSION['uid']))
{
require_once ("./includes/clientfunctions.php");
$GLOBALS['whmcsuser'] = getclientsdetails($_SESSION['uid']);

Instructions are located here
public_html/supportsuite/integrate/whmcs-integrate-howto.txt.

After all the above are done:

In logon to your Kayako install.

Create a new template group for WHMCS in Kayako, call it WHMCS or whatever you like.
In your Kayako admin panel select “Settings” then “General”
Look for “Integration Options” field, next to “WHMCS Template Group” specify your WHMCS template you created earlier.

In your WHMCS 4 or 5 install

Under “Setup->General Settings->Support
Look for Support Module and select kayako from the drop menu.
SAVE. DONE

Logon as a client and click on “My Support” or “Create Ticket” or whatever you have it named.

You are done.