Bring a large pot of lightly salted water to a boil and place a steamer insert into the pot. Break crab legs apart at joints. Place king crab legs into the steamer, cover pot, reduce heat to medium, and steam crab legs until fragrant, 6 to 8 minutes. Remove crab leg segments. When cool enough to handle, crack the crab leg shells, pick off shells, and remove meat.
Bring chicken broth to a boil in a saucepan; slowly stir in grits. Reduce heat to low, cover pan, and simmer grits until tender, about 20 minutes. Stir grits occasionally.
When grits are thickened and tender, stir Cheddar cheese and diced tomatoes with green chiles into grits until the cheese is smoothly melted.
Melt butter with olive oil in a large skillet over medium heat. When mixture begins to bubble, stir in green bell pepper and sweet onion; cook until softened, 3 to 4 minutes. Stir in garlic and cook until fragrant, about 2 minutes.
Cook and stir crab meat into bell pepper and onion until crab meat is heated through. Gently mix the cheese grits into the crab mixture; simmer 3 to 5 minutes to blend flavors, and serve.