Gene and Jude's has my favorite in the city, although Chickie's (known mostly for their beef) on 2839 S. Pulaski has solid Vienna beef wieners with natural casing (a must for a proper Chicago hot dog.) Both of these places are the minimalist style (onions, mustard, relish, optional sport pepper) of Chicago hot dog, not the drag-through-the-garden style on a poppy seed roll.
And Portillo's, despite it being a local chain, does a solid job with the dogs, too, more of the stereotypical "Chicago dog." (The only difference, I think, is they don't use the neon relish.) The char dog at the infamous Wiener Circle is one of my favorites, too. Just don't go there on a weekend night when the drunken frat ******bag crowd comes out.
Here's another vote for Hot Doug's. The duck fat fries are on Fridays and Saturdays. Their standard Chicago dog is stellar. The Game of the week is always fantastic. The veggie dog is even great. How is that possible?
Byron's also makes a pretty solid dog. There is one on Lawrence and I think another location is out there somewhere.
It used to be Demon Dogs, under the L at Fullerton. But sadly, they closed a few years ago to make way for a new platform. I'm actually pretty impressed with America's Dog. Great topping options with regional favorites from across the States...the menu will blow your mind.