Client not paying, and now sending threatening emails.

Hi there!
I'm not sure if anyone can help me or not, but basically, I completed some work for a client around two months ago now, and it's all turned quite nasty now that I've brought up payment.
The client is based in Australia, and approached me through my design page on facebook asking for some logo and branding work to be completed for the charity he works for.
I took all the details, and we were in constant contact throughout the project, initially through facebook, but then via email.
Prior to starting the work we agreed on a price, which was slightly less than I usually charge per day, however it was for a good cause and they were just starting up so I thought I would try and take some pressure off for him.
I completed the work which consisted of multiple logo options, and branding materials, and he said they were all 'amazing' so I sent the invoice to him, to which he replied saying he would pay in the next week.
Then he went quiet, and stopped responding to emails, so I continued communications via Facebook Messenger (as this way it showed the messages were being sent and read), and after the invoice being over a month overdue, and me sending four messages which he had read and ignored I messaged him one final time informing him if he wasn't in contact soon I would have to seek legal advice (this has never happened to me before).

He then replied immediately saying I was harassing him, and that he wouldn't be paying me anything as I hadn't provided goods as stated, and also said I had broken confidentiality by emailing a supplier of the charity (which I found on their website) asking if they had heard anything from them, as they seemed to have gone off the radar (obviously I haven't signed any confidentiality agreement, or given out any information which is sensitive as I have no such things!)

I have no real idea what to do next, as I have proof that all emails I sent him were perfectly friendly, granted a little short towards the end stages, but friendly nonetheless! I'm just panicking slightly as I'm pretty sure I haven't done anything wrong, and dont really know how to go about trying to receive my payment!

Any help anyone could provide would be incredibly helpful, as I'm only just starting out freelancing and this experience has shaken me a bit!

Many thanks,

Katrina