I did that ride a few years back on my concours.
Here's a wonderful sign you're greated with when you start the fun.
I actually got sick of turning after awhile. I longed for a straight section where I could rest. I think they only stock the "curves" signs up to "25 miles", because it was alot longer than 25 miles of curves.
Definitely drive south to north, going up hill.
Careful in the spring. Winter sanding of roads for snow/ice and spring rains can make for some sandy/slippery roads. If you're looking to really tear up the corners then oct/nov is perfect.
June wouldn't be too bad on the 191. Kind of hot getting to the south end if you go through phoenix or tucson. June's temperatures in the lower elevations will be in the 100's. Nights are cooler though. It'll be nice and cool once you get north of Clifton.
When I did my ride in July, I left Tucson at night when it was cooler and got a hotel in Safford and started early the next day in cooler temperatures. I rode from Safford up the 191 and then west to Pinetop
. 200 miles, and I was pooped.
The highway used to be Arizona 666, but was renamed to the 191. Hence, devils highway. Not to be confused with the other Devils Highway (El Caminiones Del Diablo) in southwestern arizona that's a dirt road that snakes along the US/Mex border.
Here are a couple of links gps "trails" of 191 on ridewithgps.com if you're using a GPS...
Just the 191
The 191 with a loop into New Mexico
The loop would be cool too. I've driven highway 78 that's just east of 191, and that's another beautiful ride. If I had to choose though, 191 would be the winner.