You can just phone them and they will "invite" you to upgrade to Platinum. I don't think there is a minimum spend anymore - they will give the card to anyone who asks for it.

If you are cheeky, you can phone up to cancel your gold card and see if the card retention will upgrade you and waive the platinum card fee for the first year. (there is no way of knowing if they will waive your fee or not, every case is different but it's worth a go)