I have some more info, gained through experimentation. ; )
There are two campaigns with the event,
, and Portobello's Game
I was able to complete Respen's Game campaign by running a few dungeon runs of Respen's Marvelous Game from the queue. These are the dungeons created two years ago. You have to complete each one at least once to get the items to complete the campaign. (which is complicated, as it randomly decides if you will do the dungeon, or the forest. I had to run the dungeon 4 times before I was lucky and got the forest!.)
Now I have the power of Lathander's Dew — Icewind Freeze! A caffeine drink that will give you a jolt to quickly regain action points!
It would seem the Portobello's Game
campaign is much more complicated to complete.
There are two parts to the campaign:
If you run Die Nation 20 times, then you get an Owl Bear Figurine, which sounds like it creates a minion that fights for you for a bit. (Or maybe it's an oddly worded companion?)
For the main part of Portabello's campaign, it looks like you need to get accolades, which can get you outfits, a gargoyle companion, and an artifact (Apocalypse Dagger, [power artifact] which deals area affect Fire damage.)
Right now I have 12 accolades, and it seems you get 4 accolades each time you run one of the weekly queues (like "Trial of Traps" this week.)
(There were two Daily quests in the new zone, perhaps Flabbergast’s daily Challenge gives 4 accolades?
I couldn't get the race to start, perhaps that gives accolades too....
From the text in the window, it looks like we can go out into the game and do things to earn accolades....
400 seems like a lot, even for a 4 week event, but I'll know more as time goes on. (I guess it's just 100 event runs.... ; )