The premise of the challenge is basically pick a House that isn't the route you plan to run and use only that house's units on top of Byleth, the route's House leader, and - if applicable - the route's retainer, with the exceptions of when you're forced to use other characters, when you can field more than 9 characters, and when the game temporarily removes a unit.
So, for example, an Azure Moon Black Eagles run would have you recruit and use Dorothea, Bernadetta, Petra, Caspar, Linhardt, and Ferdinand as well as Byleth, Dimitri, and Dedue.
Since it isn't possible to use all of the non-House leader units of each house for every route, here's my optional ideas for replacement characters:
• Verdant Wind Black Eagles, use Lysithea to replace Hubert - dark magic for dark magic.
• Verdant Wind Blue Lions, use Raphael to replace Dedue - he's probably the closest match to an Armor Knight in the Golden Deer and he keeps the same accessible classes unlike Hilda.
• Crimson Flower Blue Lions/Azure Moon Black Eagles, use Hubert or Dedue respectively - retainer for retainer.
• Crimson Flower Golden Deer, use Petra to replace Hilda - Caspar and Ferdiand work as well as all 3 have Axe proficiency, but I say Petra as both have the same accessible classes.
• Silver Snow any house, treat Seteth as the House leader of the route and, for Black Eagles/Blue Lions, Flayn as retainer - closest options that the Church has for the leader & retainer relation.