Does anyone know the driving conditions of the North Slope Road this time of year?

I don't know anything specifically about the North Slope Road, but as of Friday, there is virtually no snow below 9000' in the Uintas, but at 10000' there is about a foot or so. There is a gate to that road on the Mirror Lake Highway end so even if it is driveable for a long stretch, the gate might be closed.

Is there another way to get to North Slope then? Evanston?

You can get to the west end of the road via either Kamas or Evanston, both will get you to the same spot. If the Mirror Lake Highway gates are closed, then you have to go through Evanston. In addition to the Mirror Lake Highway gates at mile markers 14.6 and 48, there is an additional gate on the North Slope road, which branches off of the Mirror Lake Highway at about mile marker 48.

To get to the east end of the North Slope road, you have to go all the way through Mountain View, just like you're heading to Kings Peak, except that instead of taking the road to Henrys Fork, you stay on pavement for awhile and then take the road to the East Fork Blacks Fork TH. The North Slope road branches off a few miles before the TH. I don't know if there are any gates on that end or not.