They start a little before 12:30 PM but you need to get there super early to get a good spot. I was there a month ago and at 11 AM, the lotto line already stretched to the end of the block (if you're not familiar with the geography, suffice it to say it was incredibly long). If you want to just watch the show and don't enter the lotto (which I did since I'd seen it the night before), there are security/police around who don't really let you congregate before they let people start entering the lotto. It is possible to get a spot instead of letting them shunt you across the street, but it takes finagling. And because I went a month ago, before Lin announced his departure and the Tony Awards happened, it will probably be a lot busier.
Hamilton Emotional Support Group:
https://www.facebook.com/groups/hamiltonsupport/