How do I take a private car from Chiang Mai to the Laos border for a visa run?

Yes. Every time I drive to Laos I take my car across the border, because that's just how I roll.

You'll need your blue book and purple international transport permit obtained from the land transport department.

You'll need 3rd party insurance for Laos, which can be purchased at the border during business hours. Arriving outside of business hours or through a remote crossing and you'll need to purchase insurance in the nearest town.