The champions are all seperate fights. Some "champions" however are a group of mobs. Not all champions are always up, sometimes you have to wait for them to respawn... and when you don't know where they spawn, that can mean quite a bit of running around, or trying to guess their "spot" (Kinda like... "Why is this open space off the track even here? Oh, right maybe that's one of the chamipon locations...")
There even is a champion good at hiding in the orsinium dwemer dungeon (took me quite a while to find the ice atronarchs hiding spot when I was there to solo that one!) But with some effort, time and possibly looking where more experienced players look for bosses to do, you can get all those neat achievments and such!

As has been stated, the champions have nothing at all to do with the group challenge. Most of which are following a "kill all the sideshow mobs to make group boss appear" pattern. Sometimes there's also some other interaction thrown into the mix, or some specialty to the fight (like the old orsinium group challenge, where its very easy to wipe if you don't realize the flame danger pattern). And yeah, the one in the Alik'r one wants a puzzle solved where you need a second player to stand on the right plates. But none of them have -anything- to do with the rest of the dungeon. At all.