Initialize in your controller

public function action_signup()
{
    $captcha = new Captcha; // Or new Captcha('alpha'), new Captcha('math'), ...
    ...

Render captcha field in your view

<label for="captcha"><?php echo $captcha; ?></label>
<br />
<input id="captcha" name="captcha" dir="ltr" title="Key in the charaters or answer the question above" type="text">

Validate captcha

$valid = new Validate($_POST);

$valid->filter(TRUE, 'trim')
    ->rule('captcha', 'not_empty')
    ->rule('captcha', 'Captcha::valid');

Want more detail ?

Captcha docs from kohanaframework.org