Has anyone done this drive? Is it boring or are there things to do along the way to make it fun?

You might want to use www.viamichelin.com to figure out an scenic itinerary, but it looks like the most direct route is through Yellowstone which can't be too boring.

It will be scenic (and not boring) if you drive via Red Lodge, 212 (which is the Beartooth Highway), and Highway 226 (Chief Joseph Highway), and 120 into Cody.

HTTY

You can't go wrong if you follow HTTY's advice above.\
The alternative of I-90 to Laurel and then south through Bridger to Codey is not boring....well not too boring, but far from a scenic as the Yellostone route.

Beartooth is beautiful but only drivable certain months so look that up before you go.

If you just drive from the NE entrance of Yellowstone to the Chief Joseph Hwy and then down to Cody you have missed the scenic part of the Beartooth Highway. You have to drive to somewhere along the Wyoming/Montana border to really see the Beartooth Highway scenery. It is very worth it to drive that far, turn around and see it again and then go down the Chief Joseph Highway to Cody. The CJH is scenic too but not nearly as scenic as the BH.

I agree that you better check road conditions before you take that route.