Ouray, Silverton, and Telluride all have jeep tour companies that will take you over Black Bear Pass. You can also rent ATVs, which are allowed on both black Bear and Imogene Passes (but not on the highway!).
But to answer your question, no Chevy 2500, even with 4WD, is going to be able to get to the top--the road is too narrow. You might be able to drive it over Ophir Pass, but even on that you will have to do multiple turns to get around the hairpins. I've had to do that on Ophir and Cinnamon Passes even with a mid-size truck (Ford Ranger and Nissan Frontier).
Not too many years ago, some idiot drove a 3/4-ton truck up one of the jeep roads in the San Juans and couldn't turn around, so he drove out onto the tundra to the edge of a cliff. Couldn't back up, couldn't turn around, and now he was really stuck. The truck was finally taken apart and hauled out by heavy-lift helicopter. He not only got the bill for that, but he also got fined $20,000 by the Forest Service for destroying the tundra!