Send Mailchimp newsletter to List via PHP

Yes, you can. The details and examples from MailChimp are available by logging in to their control panel. Use their form fields, style your own form.

<form action='http://xxxx.xxxxlist-manage.com/subscribe' method='post'>
    <p><input type="email" value="" name="EMAIL" class="required email" id="mce-EMAIL" placeholder="enter email address"></p>
    <p><input type="submit" value="Sign Up" name="subscribe" id="mc-embedded-subscribe" class="btn"></p>
    <input type='hidden' name='u' value='xxxxxxx'>
    <input type='hidden' name='id' value='xxxxxxx'>
</form>

If you don't want to upload your template to Mailchimp and send a campaign by hitting their API, Mandrill (as @Whymarrh mentioned above in the comments) may be a good option.

Although it's meant for transactional emails (welcome, password recovery, etc), you can send to up to 1000 users at a time through SMTP. Plus you can connect your Mailchimp account to your Mandrill one in the Integrations section to track recipient activity.

My suggestion would be to install the Mandrill PHP API client, upload your template to Mandrill, hit the Mailchimp API for your user list, and then feed that into a Mandrill send-template call that you trigger through your admin panel. (Pro tip on sending mass emails: Mandrill sending to multiple people as separate message via the REST API).